Mascia et al.'s primality conjecture for polyominoes

Let $\mathcal{P}$ be a polyomino. A polyomino is prime when its inner 2-minors ideal is prime, and a zig-zag walk is the configuration referred to in the source; every prime polyomino contains no zig-zag walk.

Mascia et al.'s conjecture. $\mathcal{P}$ is prime if and only if $\mathcal{P}$ contains no zig-zag walk.

The conjecture would give a characterization of primality for polyominoes. The forward implication is known, while the converse is presented as conjectural; its general status is open.