The squarefree-triangulation conjecture for symmetric doubly-stochastic matrices

Let $S_n$ be the polytope of symmetric doubly-stochastic matrices, and let $h^*(S_n)$ denote its Ehrhart $h^*$-polynomial. A regular unimodular triangulation of $S_n$ should exist, and consequently $h^*(S_n)$ should be unimodal when $n$ is even. This would connect the combinatorial structure of $S_n$ with the unimodality of its Ehrhart series; the source gives this as a conjectural consequence of proving squarefreeness of the relevant initial terms, with no resolution stated.