Nadler–Tanaka's cosheaf conjecture for local Lagrangian cobordism categories

Let $M$ be a conical symplectic manifold with support Lagrangian $\Lambda\subset M$. For each conical open subset $U\subset M$, write $\Lag_{U\cap\Lambda}(U)$ for the corresponding stable $\infty$-category, and let $\operatorname{StCat}$ be the $\infty$-category of stable $\infty$-categories.

Nadler–Tanaka's cosheaf conjecture. The functor

$$\operatorname{Open}^+(M)\longrightarrow\operatorname{StCat},\qquad U\longmapsto\Lag_{U\cap\Lambda}(U)$$

forms a cosheaf. Thus, for every finite cover $\coprod_{\alpha\in I}U_\alpha\to M$ by conical open subsets, $\Lag_\Lambda(M)$ is the colimit of the associated Čech simplicial stable $\infty$-category whose $k$-simplices are

$$\coprod_{\underline\alpha\in I^k}\Lag_{U_{\underline\alpha}\cap\Lambda}(U_{\underline\alpha}),$$

where $U_{\underline\alpha}=U_{\alpha_1}\cap\cdots\cap U_{\alpha_k}$.

A cosheaf property would provide a local-to-global construction of $\Lag_\Lambda(M)$, avoiding the long-distance difficulties that obstruct analogous results for partially wrapped Fukaya categories. The source states this as conjectural.