The König-type conjecture for non-simple thin polyominoes

Let $\mathcal{P}$ be a non-simple thin polyomino, and let $I_{\mathcal{P}}$ be its polyomino ideal.

König-type conjecture. The ideal $I_{\mathcal{P}}$ is of König type.

The paper proves this assertion for closed path polyominoes, which form the specific class studied there. The supplied text gives no resolution of the conjecture for arbitrary non-simple thin polyominoes.