Navarra–Qureshi–Rinaldo domino-stability conjecture for Gorenstein cell ideals

Let $\mathcal{P}$ be a collection of cells, and let $K[\mathcal{P}]$ be its coordinate ring. A collection of cells is domino-stable when it has the domino-stability property defined in the source.

Domino-stability conjecture. The ring $K[\mathcal{P}]$ is Gorenstein if and only if $\mathcal{P}$ is domino-stable.

Domino stability is known to characterize palindromicity of the switching rook polynomial, and computational evidence proves the Gorenstein conclusion for domino-stable collections of rank at most $10$ and domino-stable polyominoes of rank at most $12$. The conjecture asserts the corresponding Gorenstein characterization for arbitrary collections of cells.