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:

$$tr(Ae^{B+C})≤tr(Ae^Be^C)$$ Note that if $A=I$ then this is the Golden Thompson inequality:

$$tr(e^{B+C})≤tr(e^Be^C)$$

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.