Signed-cardinality homomesy conjecture for interval-closed sets of products of two chains

Let $[m]\times[n]$ be the product of two chains, and let an interval-closed set mean a subset that is interval-closed in this poset. For $x\in[m]\times[n]$, define the signed cardinality by the parity of its rank, and for an interval-closed set $I$ let $\mathrm{sc}(I)$ be the sum of these signs over $I$. A statistic is $0$-mesic if its average on every rowmotion orbit is zero. Signed-cardinality homomesy conjecture. If $m=2$ or $m=3$, then the signed cardinality statistic is $0$-mesic under rowmotion on interval-closed sets of $[m]\times[n]$ whenever $m+n-1$ is even.

The conjecture extends homomesy results for ordinal sums of antichains and the known $[2]\times[n]$ case; the source states that its $m=2$ case is proved by the paper's Theorem 1. The $m=3$ case remains unresolved here.