Divisibility by four for even white holes in odd rectangles

Determine whether, except possibly when both odd side lengths r and c are congruent to 3 modulo 4, the number a(r,c;h) of near-perfect matchings of an odd-by-odd r×c rectangular grid with any specified even white hole h is divisible by 4.

Background

For odd r and c, the paper studies a(r,c;h), the number of near-perfect matchings of the r×c grid whose unmatched vertex is a specified hole h. It classifies exactly when this count is odd, showing that for a white hole it is odd precisely when at least one side length is 1 modulo 4, gcd(r+1,c+1)=2, and the hole has the appropriate odd position.

The paper then discusses counts modulo 4. It notes that the parity argument can be extended in some cases but leaves unresolved the stated divisibility-by-four behavior for even white holes, allowing an exception when both side lengths are 3 modulo 4.

References

We conclude with one example conjecture: Except sometimes when $r$ and $c$ are both congruent to $3$ modulo~$4$, the count $a(r,c;h)$ is a multiple of $4$ for any even white hole $h$.

— A Note on One-Hole Domino Tilings of Squares and Rectangles  (2502.05918 - Byun et al., 9 Feb 2025) in Conjecture, Section 3, “A Rectangle minus a Hole”