Root-of-unity conjecture for monodromy eigenvalues of persistent local systems

Let $K$ be a locally compact and paracompact cellular complex, let $G$ be a finite quotient of $K$, and let $\underline{\mathcal{L}}_i(G)$ be its $i$th persistent local system. Choose a cellulation of $\mathbb{S}^1$ and let $M_i$ be the invertible matrix representing the monodromy of $\underline{\mathcal{L}}_i(G)$ around that cellulation.

Root-of-unity conjecture. The eigenvalues of $M_i$ must be roots of unity.

Equivalently, some positive power of $M_i$ should be the identity, which would imply that persistent local-system classes are detected by toroidal cycles in a suitable finite cover. This would allow toroidal-cycle information at all scales to be recovered from the eigenvalues and Jordan decomposition of $M_i$; the paper does not establish the claim.