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 semi-definite and of tract one. Does there exist $C>0$ such that the following inequality

$$\big|tr(ALBL) - tr(AL^2B) \big| \leq C||L||^2 tr\big((Id-A)B\big)$$

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