Zigzag structures of simplicial complexes of type {3,4}

Let ${\cal K}$ be a simplicial complex of type $\{3,4\}$, meaning that every $(d-2)$-face is contained in $3$ or $4$ faces of dimension $d-1$. Let $(P_1,\ldots,P_t)$ be the corresponding partition of $\{1,\ldots,d\}$, and let $l_i$ denote zigzag lengths. Type-$\{3,4\}$ zigzag conjecture. The complex is not $z$-uniform if and only if the part sizes are $(\frac d2,\frac d2)$ or all even, except for the simplex; in every non-$z$-uniform case, $\gcd(l_1,l_2)=\min(l_1,l_2)$ for any two zigzag lengths. In the two specified extreme partitions, the maximum and minimum lengths are respectively $\frac{d(d+2)}2$ and $d+2$, or $3d$ and $\frac{3d}2$. For partition $\{1\},\{2,\ldots,d\}$, the complex is $BPyr(\alpha_{d-1})$. For partition $\{1\},\{2\},\ldots,\{d-2\},\{d-1,d\}$, it has the stated orbit counts, zigzag length $6d$, intersection vectors, and parity-dependent orbit sizes from the source. The supplied text gives no resolution.