Primality conjecture for weakly connected collections of cells

Let $\mathcal{P}$ be a weakly connected collection of cells, and let $I_{\mathcal{P}}$ denote its polyomino ideal. A zig-zag walk is the particular sequence of inner intervals of $\mathcal{P}$ defined in the preceding theory. Primality conjecture. The following are equivalent:

$$I_{\mathcal{P}}\text{ is prime};$$

and $\mathcal{P}$ has no zig-zag walks. Equivalently, absence of zig-zag walks in a weakly connected collection of cells characterizes the primality of its polyomino ideal. The conjecture extends the corresponding characterization from polyominoes to weakly connected collections of cells. The paper notes examples showing that weak connectivity alone does not force primality, while earlier computational results establish the condition for polyominoes consisting of at most fourteen cells and prior work proves it for certain classes such as closed paths; the general equivalence remains open.