Floquet-isospectral antisymmetric periodic potentials

Let $q>3$, let $V$ be a nonzero $q\mathbb{Z}$-periodic potential of the form

$$V=(v_1,v_2,\dots,-v_2,-v_1),$$

and let $\Delta$ denote the discrete Laplacian. Floquet-isospectral potential conjecture. There exist such potentials $V$ for which $\Delta$ and $\Delta+V$ are Floquet isospectral.

This conjecture is motivated by the specialized spectral-invariant system and by known nonzero points when the period is divisible by $4$; its general status is not established in the supplied text.