The odd axial-puncture criterion for regularity of the bi-adjacency matrix

Let $T=T_d(I)$ be a mirror symmetric region satisfying Assumption~. An axial-puncture criterion asserts that

The odd axial-puncture criterion.

$$\det Z(T)\neq 0$$

if and only if at most one axial puncture, including the top-most puncture, has odd side length.

This conjecture is based on an extensive computer search and characterizes when the bi-adjacency matrix is nonsingular.