In horde chess, the following starting position is used:
[FEN "rnbqkbnr/pppppppp/8/1PP2PP1/PPPPPPPP/PPPPPPPP/PPPPPPPP/PPPPPPPP w - - 0 1"]
The goal of White is to checkmate Black; the goal of Black is to capture all White's pieces. The main strategy for White is to try to promote a pawn; the main strategy for Black is to get their queen or rook behind White's pawns, and destroy White's forces from behind.
It naturally comes to mind that, with good play from both sides, a situation may arise in which Black manages to capture all but one of White's pawns, and the remaining pawn promotes to a queen. In this kind of endgame, White cannot win because a lone queen cannot realistically deliver checkmate (except in very special cases which are not bound to arise if Black knows what they are doing).
Therefore, Black has to force the capture of White's queen, or accept a draw. This endgame practically becomes a game of "checkmating a queen" before the 50-move rule kicks in.
So how much material is needed to force capture of a lone queen?
I guess Black's plan will begin with promoting all their remaining pawns. To start somewhere, let us assume that Black's material after promoting everything is one king plus N queens.
With a king plus 5 queens, it seems that Black has an easy win. They just need to set up a domination position, where the 5 queens attack every square on the board:
[FEN "6qk/8/4q3/3q4/2q5/6Q1/q7/8 w - - 0 1"]
No matter where the White queen moves, it will be captured on the next move. I do not have a proof, but it seems easy for Black to achieve this position while blocking checks by White.
With a king plus 4 queens, I have not found a way for Black to force the capture of White's queen. While it seems overkill to have to attack every single square of the board in order to trap a queen, maybe the queen's mobility is so large that there is no other way.
So the question is
What is the smallest N such that a Black king with N Black queens can force the capture of a lone White queen?