Solution for An Infinite Stipulation
There are two "live zones" in the game. The simpler is in the bottom left hand corner, where Black's move ...b1=B+ will allow the White king to choose between Kc1 & Kd1, and can then pace between the two squares. At the beginning, the Black rook is free to move, but following the promotion, the Black bishop blocks the rook and cannot move itself without giving mate or forcing its own capture.
On the right of the board, the Black king has a more complicated journey to from h4 to g/h8, avoiding the army of White pawns which march forward implacably like zombies in a horror movie.
The Black king must reach f7 or h7 before a White pawn occupies g6. This will happen on White's 4th move at latest, unless the Black pawn promotes, in which case it can delay indefinitely. So if Black attempts to go via f1 e.g. 1. g3+ Kh3 2. g4 Kf2 3. g5 b1=B 4. Kc/d1, then Black cannot get through e2 without interfering with the pacing White king.
An alternative route would be via f4. For example, 1. g4 b1=B+ 2. Kc/d1 g5 3. fxg5 and now Black is blocked from ever getting to f6 or h6, unless the pawn g5 moves on to g6 to finally seal f7 & h7 for good. This shows also how 1. g4? fails in general.
So the solution is 1. g3+ Kh5 2. g4+ Kh6 3. g5+ Kh7 4. fxg6+ Kg/h8 5. f6, and now to give White a 6th move there must follow 5. ...b1=B+ 6. Kd/c1. After Black's 4th & White's 6th move are chosen correctly for parity purposes, the kings just pace backwards and forwards for the correct number of moves.
Clearly, this notion could not be implemented in a straightforward Proof Game, where effectively Position A is the starting position. If a proof game exists in n moves, then it may be prefixed with a sequence of 2.0 moves involving two knight switchbacks in 16 different ways. So no unique proof game can exist in n+2.0 moves.
However, that does leave the door open for a single diagram which admits 4 proof games, each of different length. I did this, and I need to post it on the site some time...
A challenge remains to produce a composition where k, the number of moves in the first variation, is greater than the value 5.5 which appears here. It's paradoxical that a composition addressing this task is better the "fewer" the stipulations! Perhaps some more interesting by-play could be introduced as well, since this one is somewhat dry.
The Dual Diagram Proof Segment format allowed Michel Caillaud at Andernach to demonstrate a double Allumwandlung , with White & Black promotions in the same order. That is totally staggering.
 Allumwandlung (AUW) = 4 promotions, one to each of queen, rook, bishop, knight.