Let $\mathcal H(n)$ be the set of $n\times n$ Hermitian matrices, and $\mathcal S(n) \subset \mathcal H(n)$ be the subset of density matrices, i.e., $A \in \mathcal S(n)$ iff $A$ is Hermitian, positive semidefinite and of trace one. Does there exist $c>0$ such that the following inequality 

$$\big| \operatorname{tr} (ALBL) - \operatorname{tr}\left(AL^2B\right) \big| \leq c \|L\|^2 \operatorname{tr} \big((I-A)B\big)$$

holds for all $A, B \in \mathcal S(n)$ and $L \in \mathcal H(n)$?