The eigenvalue conjecture for permutation-like matrices

Let $PL_d$ denote the set of permutation-like matrices of degree $d$, and let $A\in PL_d$ be a permutation-like matrix with $d>1$. A permutation-like matrix eigenvalue conjecture. The eigenvalues of $A$ are zero or roots of unity. Further, zero is an eigenvalue if and only if $A$ is neither a permutation matrix nor the identity matrix.

This conjecture proposes that permutation-like matrices retain the root-of-unity spectral behavior of permutation matrices, apart from possible zero eigenvalues. The supplied context reports this as an open conjecture, based on the cases $d=2$ and $d=3$.