Modulo-$4$ conjecture for near-perfect matchings of odd rectangles

Let $r$ and $c$ be odd positive integers, let $a(r,c;h)$ denote the number of near-perfect matchings of the $r\times c$ grid with a specified hole $h$, and let $h$ be an even white hole.

Modulo-$4$ conjecture. Except sometimes when $r$ and $c$ are both congruent to $3$ modulo $4$, the count $a(r,c;h)$ is a multiple of $4$.

The paper presents this as an example conjecture motivated by parity and modulo-$4$ arguments, but does not specify the exceptional cases or resolve the claim.