Primality characterization for polyomino ideals without zig-zag walks

Let ${\mathcal P}$ be a polyomino. Its polyomino ideal $I_{\mathcal P}$ is the ideal associated with ${\mathcal P}$, and a zig-zag walk is a walk in ${\mathcal P}$ of the type described in the paper.

Primality conjecture. The following conditions are equivalent:

1. The polyomino ideal $I_{\mathcal P}$ is prime.
2. ${\mathcal P}$ contains no zig-zag walks.

This conjecture proposes a complete combinatorial characterization of the primality of polyomino ideals, extending the paper's results for polyominoes containing zig-zag walks, polyominoes of rank at most $14$, and grid polyominoes. Its resolution is not specified in the supplied text.