The Boundary Conjecture for doubly stochastic single eigenvalues

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices, and let a convex combination of permutation matrices mean a matrix of the form $\alpha P+(1-\alpha)Q$ with $0\leq\alpha\leq 1$. Boundary Conjecture. Every point of the boundary of $DS_n$ is an eigenvalue of a convex combination of at most two permutation matrices. This conjecture proposes a tractable description of the boundary of the doubly stochastic single eigenvalue region; it is supported by computations and by the known cases where $DS_n=PM_n$, but the paper does not establish it in general.