The center and Azumaya-locus conjecture for infinitesimal Cherednik algebras

Let $H_c$ be the infinitesimal Cherednik algebra associated with $\mathfrak{g}=\mathfrak{gl}_n$ and parameter $c$ over a field of characteristic $p$, equipped with its standard filtration. Write $\operatorname{gr} H_c$ for the associated graded algebra and $\mathfrak{Z}(A)$ for the center of an algebra $A$. For sufficiently large $p$, consider the smooth and Azumaya loci of the center $\mathfrak{Z}(H_c)$. Center and Azumaya-locus conjecture. For $p\gg 0$, one has

$$\operatorname{gr}\mathfrak{Z}(H_c)=\mathfrak{Z}(\operatorname{gr} H_c),$$

and the smooth and Azumaya loci of $\mathfrak{Z}(H_c)$ coincide. This is prompted by the analogous result for Cherednik algebras in positive characteristic, where the smooth and Azumaya loci coincide; the status of this assertion for infinitesimal Cherednik algebras is not resolved in the supplied text.