The local systems of categories embedding conjecture

Let $M$ be a manifold. Let $\mathrm{LocSysCat}(M)$ denote the $(\infty,2)$-category of local systems of categories on $M$, and let $^{2}\mathrm{Fuk}(\mathrm{T}^{*}[1]M)$ denote the $2$-category associated to the $1$-shifted cotangent bundle. Let $^{2}\mathrm{Fuk}_{M}(\mathrm{T}^{*}[1]M)$ be the subcategory whose objects are $1$-shifted Lagrangians factoring through the zero section. Local systems of categories embedding conjecture. There is a fully faithful embedding

$$\mathrm{LocSysCat}(M)\to{}^{2}\mathrm{Fuk}(\mathrm{T}^{*}[1]M).$$

Its essential image contains the objects arising from symplectic fibrations and is contained in $^{2}\mathrm{Fuk}_{M}(\mathrm{T}^{*}[1]M)$. This is motivated by the expected correspondence between symplectic fibrations and local systems of categories, but the embedding and its essential-image description remain open.