Gorenstein characterization conjecture for collections of cells

Let $\mathcal{P}$ be a collection of cells. The Gorenstein characterization conjecture asserts that the following are equivalent:

1. $K[\mathcal{P}]$ is Gorenstein;
2. the $h$-polynomial of $K[\mathcal{P}]$ is palindromic;
3. the switching rook polynomial of $\mathcal{P}$ is palindromic;
4. $\mathcal{P}$ is domino-stable.

Computations verify the claim for collections of cells of rank at most $10$ and polyominoes of rank at most $12$. The conjecture would characterize the Gorenstein property, including for coordinate rings that are not domains, through palindromicity and domino-stability.