The non-dualizable quasi-monodromic categorical representation conjecture

Let $G$ be a group, let $\Shv(G)^{\on{q-mon}}$ denote the quasi-monodromic sheaf category, and write $G\mmod$ for the category of categorical representations of $G$. Let

$$\Shv(G)^{\on{q-mon}}\commod(\DGCat) \overset{\bi}\to G\mmod^{\on{q-mon}}\to G\mmod$$

be the indicated composition, and let $\bi:\DGCat\to\AGCat$ be the functor sending a DG category to its underlying object of $\AGCat$. The non-dualizable quasi-monodromic categorical representation conjecture. The essential image of the composition consists of those objects of $G\mmod$ for which the underlying object of $\AGCat$ lies in the essential image of $\bi:\DGCat\to\AGCat$. This conjecturally removes the dualizability assumption from the preceding theorem and would give a direct-sum decomposition into quasi-monodromic pieces for every such categorical representation.