The Type III stochastic-matrix graph conjecture

Let $A$ be a stochastic matrix of order $n$ whose characteristic polynomial is Type III:

$$f_{\alpha}(t)=t^y(t^q-(1-\alpha))^d-\alpha^d,$$

where $n=qd+y$, $1\leq y\leq q-1$, and $d\geq 2$. Type III stochastic-matrix graph conjecture. Then $A$ is permutationally similar to a matrix whose directed graph $\Gamma$ satisfies the hypothesis of Proposition 3.

This conjecture proposes that every stochastic matrix with a Type III characteristic polynomial has, after a simultaneous permutation of rows and columns, the graph structure described by the cited proposition. The supplied text gives no resolution or further context, so its status is open.