Levelness conjecture for simple thin path polyominoes with singleton maximal intervals

Let $\mathcal{P}$ be a simple thin polyomino, meaning a path polyomino with no interval containing more than two cells, and suppose that every maximal interval consists of a single cell. Let $S/I_{\mathcal{P}}$ be the quotient by the associated polyomino ideal. Levelness conjecture. The ring $S/I_{\mathcal{P}}$ is level. The conjecture is motivated by the fact that the path polyominoes covered by the preceding theorem have at least one single cell in every interval, together with computational evidence; its general validity remains open.