5 problems

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

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

Mascia–Rinaldo–Romeo conjecture. The following conditions are equivalent:

Hibi–Qureshi conjecture. The ideal $I_{\mathcal{P}}$ arises as the toric ideal of the edge ring of a finite simple graph if and only if $\mathcal{P}$ is simple.

Cell-count height conjecture. The height of every polyomino ideal $I_{\mathcal P}$ is the number of cells of ${\mathcal P}$.