The trace conjecture for perverse schobers

Let $X$ be an algebraic variety or stack, and let $\mathbb{L}\subset T^*X$ be a conic Lagrangian. Write $\mathfrak{L}X$ for the loop space, and let $\Lambda(\mathbb{L})\subset T^*\mathfrak{L}X$ be the Lagrangian determined by $\mathbb{L}$. The categorical trace of the 2-category of perverse schobers on $X$ with singular support in $\mathbb{L}$ should be equivalent, as an $S^1$-category, to the category of sheaves on $\mathfrak{L}X$ with singular support in $\Lambda(\mathbb{L})$:

$${\sf Tr}(2\mathsf{Perv}_{\mathbb{L}}(X))\simeq \mathsf{Sh}_{\Lambda(\mathbb{L})}(\mathfrak{L}X).$$

This conjecture proposes an A-model description of the categorical trace of perverse schobers, motivated by 3-dimensional mirror symmetry and the expected relationship between symplectic and coherent 2-categories. The Lagrangian $\Lambda(\mathbb{L})$ is only approached or constructed in special cases, so the conjecture remains open.