Mascia–Rinaldo–Romeo primality conjecture for polyomino ideals

Let $\mathcal{P}$ be a polyomino, and let $I_{\mathcal{P}}$ be its polyomino ideal. A zig–zag walk is the combinatorial configuration defined earlier in the source. The rank of $\mathcal{P}$ is the number of its cells.

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

1. $I_{\mathcal{P}}$ is a prime ideal.
2. $\mathcal{P}$ contains no zig–zag walk.

The equivalence is known for polyominoes of rank at most $14$, based on systematic enumeration and computation. The conjecture asserts that the zig–zag obstruction is the only obstruction to primality in arbitrary rank.