Zig-zag collection containment in every non-prime collection of cells

Prove that every collection of cells P whose inner 2-minor ideal I_P is not prime contains a zig-zag collection as a subcollection, namely a union of cell intervals supported by a sequence of inner intervals that intersect consecutively at single vertices and satisfy the edge-interval alignment conditions specified in Definition 3.1.

Background

The authors introduce zig-zag collections, a structured class of non-prime collections built from inner intervals that intersect at single vertices in a cyclic pattern. They use these to analyze Cohen–Macaulayness and Hilbert–Poincaré series of closed paths in the non-prime setting.

They explicitly refer to a conjecture (from prior work) positing that every non-prime collection of cells contains a zig-zag collection, suggesting a decomposition approach that could systematize the paper of non-prime cases.