Velocity-scaling conjecture for quantum walks in periodic fields

Let $t\in(0,1)$ and $n\in\mathord{\mathbb N}$. Let $\mathcal{D}_n$ be the corresponding local $n$-periodic field, and let $U_{t,n}$ be the associated quantum walk. Its velocity is denoted by $v_{U_{t,n}}$. Velocity-scaling conjecture. The velocity of the quantum walk scales like $t^n$, namely

$$v_{U_{t,n}}=\mathcal{O}(t^n).$$

This conjecture proposes that periodic fields slow quantum walks exponentially in their period and would extend the velocity bound beyond the restriction $t\in(0,1/4)$ in the main theorem. The stated numerical experiments support the scaling for the tested parameters, but the conjecture is not resolved in the supplied text.