The two-value criterion for generalized distance-matrix cospectrality

Let $G$ and $H$ be graphs, and suppose that a single similarity matrix establishes cospectrality of their generalized distance matrices for two distinct values of $q$, neither of which is $0$ or $1$. The two-value criterion. If graphs are cospectral for two different $q$ values (not $0$ or $1$) using the same similarity matrix, then they are cospectral for all values of $q$. This would give a way to certify cospectrality for every value of $q$ from cospectrality at just two nontrivial values, and is posed as a question for future work; no resolution is provided in the source.