The switching-rook-polynomial conjecture for polyominoes

Let $\mathcal{P}$ be a collection of cells, let $\mathbb{K}[\mathcal{P}]$ denote its coordinate ring, and let $r(\mathcal{P})$ be its rook number. Define the switching rook polynomial by

$$\widetilde{r}_{\mathcal{P}}(t):=\sum_{j=0}^{r(\mathcal{P})}\widetilde{r}_j(\mathcal{P})t^j,$$

where $\widetilde{r}_0(\mathcal{P})=1$ and $\widetilde{r}_j(\mathcal{P})$ is the number of equivalence classes of $j$-rook configurations under the switching equivalence relation.

Switching-rook-polynomial conjecture. The $h$-polynomial of $\mathbb{K}[\mathcal{P}]$ is equal to $\widetilde{r}_{\mathcal{P}}(t)$, and its regularity is equal to $r(\mathcal{P})$.

This conjecture combines proposed formulas for the Hilbert series and regularity of coordinate rings of collections of cells. The formulas agree with known results in the thin case, but the assertion for arbitrary collections of cells remains open.