Microlocal endomorphism conjecture for singular support

Let $X$ be a smooth rigid analytic variety, and let $\mathrm{F}\in D^{(b)}_{zc}(X)$. Here $SS(\mathcal{F})$ denotes the singular support and $\operatorname{supp}\,\mu hom(\mathcal{F},\mathcal{F})$ the support of its microlocal endomorphism object.

Microlocal endomorphism conjecture.

$$SS(\mathcal{F})=\operatorname{supp}\,\mu hom(\mathcal{F},\mathcal{F}).$$

This identity would in particular give positive answers to the questions concerning smooth pullback and analyticity of singular support. The source does not state a general resolution status.