Nadler–Tanaka's constructible-sheaf model for cotangent bundles

Let $X$ be a space, let $M=T^*X$, let $\Lambda=X$ be the zero section, and set

$$\cE=\operatorname{End}_{\Lag_{pt}(pt)}(pt).$$

Let $\operatorname{Perf}(X,\cE)$ denote the stable $\infty$-category of perfect constructible sheaves of $\cE$-modules on $X$.

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

$$\operatorname{Perf}(X,\cE)\simeq \Lag_X(T^*X)$$

which sends the constant sheaf on a point $x\in X$ to the conormal Lagrangian $T_x^*X\subset T^*X$.

This reformulates the cotangent-bundle conjecture in sheaf-theoretic terms and connects Lagrangian cobordisms with constructible and microlocal sheaf theory. The source states it as equivalent to the preceding cotangent-bundle conjecture and supplies no proof.