The microlocal schober model for 2-categorical category O

Let $(X\supset\mathbb L)$ be a sectorial conical symplectic resolution, and let $\mathbb L$ denote its distinguished Lagrangian. Write $2\mathcal O_{\mathsf A}$ and $2\mathcal O_{\mathsf B}$ for the A- and B-model 2-categories, with canonical generators $\mathcal P$ and $\mathcal S$, respectively. Let $\mathsf{HP}$ denote periodic cyclic homology, and let $\mathcal O_{\mathbb Z/2}$ be the corresponding $\mathbb Z/2$-graded category $\mathcal O$, with projective and simple objects $P$ and $L$. Microlocal schober conjecture. There is an associated pair of 2-categories

$$2\mathcal O_{\mathsf A}:=\mu\mathsf{PervCat}(\mathbb L),\qquad 2\mathcal O_{\mathsf B}:=\mu\mathsf{CohCat}(\mathbb L),$$

of microlocal perverse and coherent schobers on $\mathbb L$, and

$$\mathsf{HP}(\mathsf{end}_{2\mathcal O_{\mathsf A}}(\mathcal P))\simeq \mathsf{end}_{\mathcal O_{\mathbb Z/2}}(P),\qquad \mathsf{HP}(\mathsf{end}_{2\mathcal O_{\mathsf B}}(\mathcal S))\simeq \mathsf{end}_{\mathcal O_{\mathbb Z/2}}(L).$$

Thus periodic cyclic homology recovers the projectives and simples of the $\mathbb Z/2$-graded category $\mathcal O$. The statement is presented as a prediction for the 2-categorical origins of category $\mathcal O$; the surrounding discussion indicates that these 2-categories are richer than the usual 1-category, while their full construction and the conjectured identifications remain open.