For three symmetric positive semidefinite matrices $A, B,C$, I am trying to figure out if the following inequality holds, at least in some cases:

$$ \operatorname{tr} \left( A e^{B+C} \right) \leq \operatorname{tr} \left( A e^B e^C \right) $$

Note that if $A=I$ then this is the [Golden-Thompson inequality][1]:

$$\operatorname{tr} \left( e^{B+C} \right) \leq \operatorname{tr} \left( e^B e^C \right)$$

I wonder if the first inequality is true or if adding certain assumptions (such as $A≼I$ i.e. $I−A$ is positive semidefinite) will suffice to prove it. I have tried searching for similar inequalities with no luck.


  [1]: https://en.wikipedia.org/wiki/Golden%E2%80%93Thompson_inequality