The logarithmic commutator inequality with sharp constant one

For every pair of positive definite complex matrices $A,B$ with
$\operatorname{tr}(A+B)=1$, we prove
$$\|[B,\log(A+B)]\|_1\le-\operatorname{tr}(A)\log\operatorname{tr}(A)-\operatorname{tr}(B)\log\operatorname{tr}(B).$$
The proof uses Frenkel's integral representation of quantum relative entropy,
in its positive-part form given by Hirche and Tomamichel. The key estimate is
that $\operatorname{tr}(\rho-\gamma e^{itH}\sigma e^{-itH})_+$ is
$\|H\|_\infty$-Lipschitz in $t$, uniformly for $\gamma\ge1$ and for all matrix
sizes. A positive integral of these estimates has total weight exactly the
binary entropy. A strictly positive definite $2\times2$ family from earlier
findings proves that no coefficient smaller than one is possible.