Let $f:Y\to X$ be a morphism of smooth varieties (complex analytic or algebraic as you wish). We have $$ T^* Y \overset{f_d}{\longleftarrow} Y\times_X T^*X \overset{f_\pi}{\longrightarrow} T^* X $$ and, for any holonomic $\mathcal{D}_X$-module, we have the estimate $$ Ch(Lf^* \mathcal{M}) \subset f_d f_\pi^{-1} Ch(\mathcal{M} ). $$

We know this is an equality when $f$ is non-characteristic for $\mathcal{M}$ meaning that $$ f_d^{-1}(T^*_Y Y) \cap f_\pi^{-1}Ch(\mathcal{M}) \subset Y\times_X T^*_X X $$

Question Is this an equality when $f$ is the blow-up of $X$ along a smooth sub-variety $A$?

I have checked this in a few simple cases but I'm having trouble proving it in full generality.