The Perfect–Mirsky conjecture for dimensions 6 through 11

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices, and let $PM_n$ be the union of the convex hulls of the $k$-th roots of unity for $k\leq n$. Perfect–Mirsky conjecture for $6\leq n\leq11$.

$$DS_n=PM_n\qquad\text{for }n=6,7,8,9,10,11.$$

This is a finite-dimensional strengthening of the general Perfect–Mirsky equality, motivated by the authors’ computations; the paper presents it as conjectural rather than proved.