Zigzag structure of direct products of polygons

Let $C_p$ and $C_q$ be polygons, and set

$$t=\gcd(p,q),\qquad s=\frac{pq}{t^2}.$$

Write $z$ for the multiset of zigzag lengths and multiplicities, and $Int$ for intersection vectors. Product-of-polygons conjecture. For the direct product $C_p\times C_q$: if $p,q$ are both even, then $z=(2ts)^{6t}$ with $Int=(0,2s)^t$ for every zigzag; if exactly one of $p,q$ is odd, then $z=(2ts)^{6t}$, with $Int=(0,s)^{2t}$ for $4t$ zigzags and $Int=(s,s)^t$ for the remaining $2t$ zigzags; and if $p,q$ are both odd, then $z=(2ts)^{2t},(4ts)^{2t}$, with $Int=(s,s)^t$ for zigzags of length $2ts$ and $Int=(2s,2s)^t$ for zigzags of length $4ts$. The conjecture was checked for $p,q\leq 15$; its general validity remains unresolved in the supplied text.