Perron eigenvalue conjecture for the component ${\cal A}^{*}_m$

Let $m\geq 2$, let ${\cal D}^{*}_m$ be the relevant digraph, and let ${\cal A}^{*}_m$ be its distinguished component. The maximum eigenvalue of a digraph means the largest eigenvalue of its adjacency matrix, and a maximum-modulus eigenvalue is an eigenvalue whose absolute value is maximal. Perron eigenvalue conjecture. The maximum eigenvalue of ${\cal A}^{*}_m$ is the unique maximum-modulus eigenvalue of ${\cal D}^{*}_m$.

This claim is motivated by computations for $m\leq 12$, which indicate that the maximum eigenvalue is simple, unique in modulus, and attached to ${\cal A}^{*}_m$. The source presents the assertion as an unresolved computationally observed property.