All-order trace derivative positivity conjecture

Let $k$ be a positive integer, let $f:(a,b)\to\mathbb{R}$ have non-negative $k$th derivative in the distributional sense, and let $A,B$ be $n\times n$ matrices. For odd $k$, additionally assume that $B$ is positive semidefinite. Define

$$t\mapsto \operatorname{tr} f(A+tB).$$

All-order trace derivative positivity conjecture. The function $t\mapsto\operatorname{tr} f(A+tB)$ has non-negative $k$th derivative.

The paper proves the analogous assertion for fourth derivatives without a positivity assumption on $B$, and for third derivatives when $B$ is positive semidefinite. The conjecture extends these results to every positive integer order; the general case remains open.