Nadler–Tanaka's stratified constructible-sheaf conjecture

Let $X$ be Whitney stratified by strata $X_\alpha$, $\alpha\in A$, set $M=T^*X$, and let

$$\Lambda=\coprod_{\alpha\in A}T^*_{X_\alpha}X$$

be the union of their conormals. Set $\cE=\operatorname{End}_{\Lag_{pt}(pt)}(pt)$, and let $\operatorname{Perf}_A(X,\cE)$ denote the stable $\infty$-category of perfect $A$-constructible sheaves of $\cE$-modules.

Nadler–Tanaka's stratified constructible-sheaf conjecture. There is a canonical equivalence of stable $\infty$-categories

$$\operatorname{Perf}_A(X,\cE)\simeq \Lag_\Lambda(T^*X).$$

This extends the zero-section case from ordinary constructible sheaves to sheaves constructible with respect to a Whitney stratification. The source relates the claim to expected descriptions of Fukaya categories but gives no resolution.