Trace-norm commutator conjecture for matrix functions

Let $A$ and $B$ be positive semidefinite $d\times d$ matrices with $\operatorname{Tr}A=a$ and $\operatorname{Tr}B=b$. For a function $f:\mathbb{R}\to\mathbb{R}$, define

$$F(x)=\int_0^x f(y)\,dy.$$

Trace-norm commutator conjecture. For certain functions $f$ still to be determined, there exists a constant $c$ independent of $d$ such that

$$\lVert[B,f(A+B)]\rVert_1\leq c\bigl(F(a+b)-F(a)-F(b)\bigr).$$

This is presented as a natural generalisation of a previously discussed trace-norm inequality for commutators. The admissible class of functions $f$ and the optimal constant are left open in the source.