Hillar's commuting-minimizers conjecture for the Bessis–Moussa–Villani problem

Let $m>k>0$ be positive integers. Let $A$ and $B$ be positive semidefinite matrices of norm $1$ that minimize the trace coefficient $\operatorname{Tr}[S_{m,k}(A,B)]$ and satisfy the Euler–Lagrange equations associated with this minimization.

Hillar's commuting-minimizers conjecture. Such matrices $A$ and $B$ commute.

The conjecture is proposed as a strategy for proving the Bessis–Moussa–Villani trace conjecture: commutation would reduce the relevant trace expressions to the commuting case. The paper presents the Euler–Lagrange equations and proves related extremal results, but does not establish this commutation claim in general.