The Lieb–Seiringer BMV positivity conjecture for positive semidefinite matrices

Let $A$ and $B$ be positive semidefinite matrices of the same size, and let $m$ and $r$ be integers satisfying

$$m\geq r\geq 0.$$

Consider the coefficient of $t^r$ in

$$\mathsf{trace}((A+tB)^m).$$

Lieb–Seiringer conjecture. This coefficient is non-negative.

This is the version of the BMV positivity question stated by Lieb and Seiringer and is a stronger positive-semidefinite-matrix formulation than the symmetric-matrix variant. The source context identifies the result as a question, while the supplied status evidence records that it was proved by Stahl and simplified by Eremenko.