There is no link. This is just a theoretical proposal.
Variant: Four Player Antichess

Maybe not in the near future. And maybe never, if the proposed variant does not provide an interesting gameplay. I can imagine that the playing experience would be pretty chaotic, with pieces captured right and left on almost every move.

It's very interesting idea, especially for players like me, who are big fan of both antichess and FFA.
But I'm not sure whether we should remove under-promotion (or promote to king) rule or not, because it has a great meaning in antichess (as you know, tricky endgame is the main part of this game for advanced players, and one promotion often decides the whole game).
Also we have to consider more seriously about draw offer rules this time (it's easy to guess we'll have many K vs K vs K endgame or something like that).

@34844874497 I have not much experience of playing antichess, but my experience with 4-player FFA shows that the ability to calculate deep tactics is severely limited, when 3 other players are making moves between my moves. I expect something similar to happen with 4-player antichess. What is different is that the last two survivors of the antichess variant will be less interested about the outcome of their duel, since they will be competing for the paltry 3rd place of the ranking. I guess that both would prefer to end the game as soon as possible, and not engage too much in the trickiness of their endgame, since they are both going to lose rating points anyway (most probably).

As far as I know, when a promotion happens in antichess, promotes to a king is the most common choice, and queen is usually the worst and most likely to lose a game (of course it depends on a situation though). It means that, if you come to a situation that you have nothing but a pawn (it's very common in antichess) your destiny fully depends on whether you can under-promotion or not. I think it affects to 1st and 2nd places too.
Anyway, what's more important thing is draw rules. how can we end king endgames and similar ones? It often decided by flag or draw agreement in antichess but we have 15secs delay and no draw offer button now. And really tricky part is, while simple K vs K is a dead draw, K vs K vs K can be decisive because cooperated 2Ks can defeat 1K. While simple K vs R is a dead lost for K, K vs K vs R is not that clear.
There are tons of examples, but it can easily be understood why that happens. Both FFA and antichess has very difficult and tricky endgame theories, and what you suggest is to combine them. On the other hand, those endgames are just boring for people who don't understand those theories, and maybe some of such endgames are just theoretical draws.
After all, there are many expected problems, and I hope we will have some experiments to start real arguments.

Anti-checkers (Giveaway checkers) is a very interesting game. At least it was fun to play it when I was a schoolboy. "Anti 4 Player Chess" may be interesting too, just the server must indicate (highlight) all possible captures, otherwise it is easy to miss some capture on this huge board.

I understand that 50 move rule is rarely used in FFA now, but I expect that will happen almost every game in this 4-player antichess, especially at high rated games. I'm ok with that but I'm afraid most of people will feel boring and regard it as a waste of time.
We don't even know exactly how often that happens yet, so we need some tests anyway.
@dirtguylawra
I thought about that too. If we bring this variant to teams mode, I think most of this "endless endgame" issues will be easily solved.

K vs K vs K can be decisive because cooperated 2Ks can defeat 1K.
You mean that the two cooperated kings can force the single king to capture both of them, or one of them?

Yes. In normal antichess, 2Ks vs K is usually a win for 2Ks because they can easily set a zugzwang.
Example: White Ke5, Ke6 Black Kh8
If it's white to move, 1.K5f5 Kg8 2.Kf7 Kxf7 3.Kf6 1-0. Black to move, 1...Kh7 2.Kf4! Kh8 3.Kff5 1-0.
A similar theory can happen in K vs K vs K, but not sure because two of them must cooperate very accurately... oh wait, it's a very good puzzle itself to try to defeat in K vs K vs K under "healthy alliance" condition (cooperating 2 kings can't get too close)...
But yeah, now I agree with your suggestion that says we should remove under-promotion rule. As I realized one of the "simplest" looking king endgame is that difficult to understand, I think it's just overwhelming human capacity to decide what's the best piece to promote in complicated situations.

I don't think that it is possible for two cooperating players to force the enemy King to capture both of their Kings. Maybe it is possible to force him capture one King, and then tie with the other. But this scenario makes the cooperation not very profitable. The cooperating players will have to abandon the chance for the 2nd place, tied with the other two players (assuming that the missing forth player has already won the 1st place), for one of the two equiprobable possibilities bellow:
a) the 2nd place without tie
b) the 3nd place, tied with the enemy player
Statistically it is still profitable, but not that much. And the third player will have the pleasure to decide whom of the two partners to benefit and whom to harm.
My suggestion to remove underpromotions is mostly practical. It has not been implemented for the standard version of the game, and it is possible that it'll never be due to the low demand for the feature. I don't think that the implementation is trivial, since it requires UI development. On the contrary the antichess variant requires "only" coding the antichess rules on the existing UI, which seems not particularly difficult. But if the implementation of underpromotions happens some day, there would be no reason not to enabled it for the antichess variant!

I think it probably would be really fun, but it doesn't exist yet so I don't really know. I hope they make this variant on chess.com, and I would play it! I know the gameplay would probably be crazy and wild though!

I think if cooperating players somehow managed to make a position like following one, there's a good chance to force the other player to get the bottom (to make it simple, I use normal 8x8 board this time).
Red Kf6, Blue Kc6, Yellow Ke3, Blue and Yellow are cooperating to defeat Red, Red to move.
1.Kf7 Kc7 Ke4 2.Kg7 Kd7 Kf4 3.Kh8 Ke8 Kg5
And now Red can't avoid getting the last place but has a choice to decide who get 2nd or 3rd place.
But if cooperating kings are separated too far, like in similar situation that is almost the same but Yellow K is on h3 at starting position, if he wants, Red can share the 3rd places by playing
1.Ke7!? Kd6 Kg4 2.Kxd6 and Yellow is not in time.
As you mentioned, there's a doubt that whether players can have enough motivation to cooperate since they can't beat the other and draw them at the same time (it's theoretically impossible according to the rules).

Red Kf6, Blue Kc6, Yellow Ke3, Blue and Yellow are cooperating to defeat Red, Red to move.
1.Kf7 Kc7 Ke4 2.Kg7 Kd7 Kf4 3.Kh8 Ke8 Kg5
And now Red can't avoid getting the last place but has a choice to decide who get 2nd or 3rd place.
Excellent! I had created the same final position in my mind, and I wrongly concluded that Red is not forced to capture both Kings, but he is!

@NelsonMoore 4-player Crazyhouse has already been proposed and discussed.
About gray pieces, why do you think it's important to be captured compulsory at a lower priority than alive pieces?
It would be a simple variant, with almost the same rules as those for standard antichess. Lets review these rules (copy-pasted from Wikipedia).
The rules of antichess are the same as those for standard chess, except for the following special rules:
• Capturing is compulsory.
• When more than one capture is available, the player may choose.
• The king has no royal power and accordingly:
◦ it may be captured as any other piece;
◦ there is no check or checkmate;
◦ there is no castling;
• Stalemate is a win for the stalemated player.
• A player wins by losing all his pieces, or being stalemated.
A point system like the one used for FFA 4-player chess will not be needed here. The final rankings will be determined be the order by which the players achieve the goal of losing all their pieces, or becoming stalemated. In other words, the first player who will lose all his pieces, or become stalemated, takes the first place. The second player takes the second place. The third player takes the third place. The last surviving player takes the fourth place.
Resigning, timing-out or disconnecting will award the player the fourth place. If later another player do any of these, he will get the third place. The next one will take the second place, and the game will end with the last surviving player as the winner. The pieces of any removed players will remain on the board as dead (gray) pieces, but their capture will still be compulsory for the remaining players.
Note: In standard antichess it is allowed to promote pawns to Kings. To keep things simple and familiar we could discard this rule, and adopt the 4-player FFA rule of forced pawn promotions to Queens on each player's 8th rank (at the center of the board).
What do you think? Opinions are welcome!