Chess on a Klein Bottle

Today I discovered this 2006 article by Timothy Chow in The Mathematical Intelligencer that introduces a variant of chess played on a Klein bottle.

In Chow's game, the surface is constructed in the canonical sense with an inversion of direction when the and files are joined. Tangentially, this 2013 paper discusses the Bishop Independence problem on a couple of topologies and includes a nice visual on how the bishop moves on the Klein bottle (the paper itself is an interesting read as well!).

I found the two puzzles in Chow's article to be approachable, but positions quickly become gnarly to analyze with more pieces on the board. Here's an example, with White to move:

In a standard chess game White has mate in 1 with , but this is no longer the case in the Klein bottle variant as Black escapes to . Does White still have a forcing sequence of moves leading to checkmate in this position?

I'm still working on computing the move sequences, so I can't give a definitive answer at this time (intuition says yes; the White knight and rook are on very active squares). In the meantime, don't hesitate to reach out with a solution if you have one!

-Aaron