Non-periodicity conjecture for toroidal grids

Let $n,m \geq 3$ with $(n,m) \neq (4,4)$. A toroidal $(n,m)$-grid is the toroidal grid with parameters $n$ and $m$, and periodic means periodic under the discrete-time quantum walk at some time $\tau$. Non-periodicity conjecture. The toroidal $(n,m)$-grid is not periodic at any time $\tau$. Consequently, it has no perfect state transfer at any time $\tau$. The conjecture concerns the apparent rarity of perfect state transfer in orientable maps: the authors found it in toroidal grids with $n=1,2$ but nowhere else in their computations, and periodicity would be necessary for perfect state transfer.