Classification conjecture for parameterized families of 1+1d gapped systems

Let $\mathcal{C}$ be a fusion category, let $\mathcal{M}$ be a $\mathcal{C}$-module category, and denote the corresponding gapped phase by $\mathcal{T}_{\mathcal{M}}^{\mathcal{C}}$. Let $\operatorname{Fun}_{\mathcal{C}}(\mathcal{M},\mathcal{M})^{\mathrm{inv}}$ be the group of invertible objects in the category of $\mathcal{C}$-module endofunctors of $\mathcal{M}$. The classification conjecture. $S^1$-parameterized families of 1+1d $\mathcal{C}$-symmetric gapped systems in the phase $\mathcal{T}_{\mathcal{M}}^{\mathcal{C}}$ are classified by the group $\operatorname{Fun}_{\mathcal{C}}(\mathcal{M},\mathcal{M})^{\mathrm{inv}}$. This generalizes the classification of parameterized SPT phases to arbitrary module-category-labeled gapped phases; the paper supports it with general arguments and examples, but does not establish it in general.