The height–rank conjecture for non-simple polyominoes

Let $\mathcal{P}$ be a non-simple polyomino, let $I_{\mathcal{P}}$ be its polyomino ideal, and let $\operatorname{rank}(\mathcal{P})$ denote the rank of $\mathcal{P}$.

Height–rank conjecture. One should have

$$\operatorname{ht}(I_{\mathcal{P}})=\operatorname{rank}(\mathcal{P}).$$

The equality is known for simple polyominoes, while the paper proves it for closed path polyominoes and conjectures it for all non-simple polyominoes.