Nadler–Tanaka's cotangent-bundle conjecture for Lagrangian cobordisms

Let $X$ be connected, set $M=T^*X$ and let $\Lambda=X$ be the zero section. Choose $x\in X$ and write $T_x^*X\subset T^*X$ for the corresponding conormal Lagrangian. Set

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

For an $E_1$-algebra $A$, write $A\operatorname{Perf}$ for its stable $\infty$-category of perfect modules, and let $\Omega_xX_* = \Omega_xX\amalg\{*\}$.

Nadler–Tanaka's cotangent-bundle conjecture. The object $T_x^*X\in\Lag_X(T^*X)$ is a generator; every object is a finite colimit of objects $T_x^*X[k]$, $k\in\mathbb Z$; and

$$\operatorname{End}_{\Lag_X(T^*X)}(T_x^*X)\simeq \Omega_xX_*\wedge\cE.$$

Consequently,

$$\Lag_X(T^*X)\simeq (\Omega_xX_*\wedge\cE)\operatorname{Perf}.$$

This is the Lagrangian-cobordism analogue of the expected description of the wrapped Fukaya category of a cotangent bundle by based-loop-space chains. The source treats the assertion as conjectural.