Dimension-free trace commutator inequality

Let $A$ and $B$ be positive definite matrices satisfying $\operatorname{Tr}(A+B)=1$. Trace commutator conjecture. There exists a constant $c$, independent of the dimensions of $A$ and $B$, such that

$$\lVert[B,\log(A+B)]\rVert_1\leq c\bigl(-\operatorname{Tr}A\log\operatorname{Tr}A-\operatorname{Tr}B\log\operatorname{Tr}B\bigr).$$

The source says that numerical work suggests the sharper value $c=1$, and gives a reduction to the corresponding inequality for arbitrary positive semidefinite matrices; neither the existence claim nor the sharp constant is resolved there.