The rook polynomial characterization of thin polyominoes

Let ${\mathbb P}$ be a polyomino. Denote by $r_{\mathbb P}(t)$ its rook polynomial and by $h(t)$ the polynomial appearing in the Hilbert series of its associated polyomino ring.

Thinness conjecture. ${\mathbb P}$ is thin if and only if

$$r_{\mathbb P}(t)=h(t).$$

This conjecture proposes that equality between the rook polynomial and the relevant Hilbert-series polynomial characterizes thin polyominoes, including among thin polyominoes that are not simple. The supplied text gives no evidence that the conjecture has been resolved.