Extension of geometrically constructed wrapped sheaves to the cotangent bundle

Let $M$ be a manifold and let $\Lambda\subseteq T^\ast M$ be a singular isotropic. Nadler's conic cosheaf of wrapped microlocal sheaves is a functor

$$\operatorname{\mu Sh}_\Lambda^w:\operatorname{Op}_{T^\ast M}\rightarrow \operatorname{st}_\omega.$$

Its restriction to the zero section is the cosheaf $\operatorname{Sh}_\Lambda^c$ of wrapped sheaves, while the geometrically constructed wrapped-sheaf cosheaf is denoted by $\operatorname{\mathfrak{w} sh}_\Lambda$.

Extension conjecture. The construction $\operatorname{\mathfrak{w} sh}_\Lambda$ should extend from the zero section to the cotangent bundle $T^\ast M$.

The paper explains that Corollary of its main theorem identifies the two wrapped-sheaf cosheaves on the zero section. The proposed extension would provide a geometrically constructed counterpart to Nadler's categorical cosheaf of wrapped microlocal sheaves.