Rinaldo–Romeo thinness conjecture for rook polynomials of polyominoes

Let $\mathcal{P}$ be a polyomino. Write $r_{\mathcal{P}}(t)$ for its rook polynomial, $r(\mathcal{P})$ for its rook number, $h_{K[\mathcal{P}]}(t)$ for the $h$-polynomial of its coordinate ring, and $\operatorname{reg} K[\mathcal{P}]$ for its regularity. A polyomino is thin when it has the thinness property defined in the source.

Rinaldo–Romeo conjecture. The polyomino $\mathcal{P}$ is thin if and only if

$$r_{\mathcal{P}}(t)=h_{K[\mathcal{P}]}(t).$$

Moreover, the source asks whether

$$\operatorname{reg} K[\mathcal{P}]=r(\mathcal{P}).$$

For simple thin polyominoes, the equality of the $h$-polynomial with the rook polynomial and the equality of regularity with the rook number are known. The conjecture asks whether the polynomial characterization extends to all polyominoes and whether the regularity equality holds in general.