Non-abelian cohomology conjecture for parameterized 1+1d gapped systems

Let $X$ be a sufficiently nice parameter space, such as a CW complex, let $\mathcal{C}$ be a fusion category, and let $\mathcal{M}$ be a fixed $\mathcal{C}$-module category with corresponding gapped phase $\mathcal{T}_{\mathcal{M}}^{\mathcal{C}}$. Let $\underline{\operatorname{Fun}}_{\mathcal{C}}(\mathcal{M},\mathcal{M})^{\mathrm{inv}}$ be the 2-group of invertible objects and morphisms in the category of $\mathcal{C}$-module endofunctors of $\mathcal{M}$, and let $B\underline{\operatorname{Fun}}_{\mathcal{C}}(\mathcal{M},\mathcal{M})^{\mathrm{inv}}$ denote its classifying space. The non-abelian cohomology conjecture. $X$-parameterized families of 1+1d gapped systems in $\mathcal{T}_{\mathcal{M}}^{\mathcal{C}}$ are classified by

$$[X,B\underline{\operatorname{Fun}}_{\mathcal{C}}(\mathcal{M},\mathcal{M})^{\mathrm{inv}}]$$

of homotopy classes of maps, equivalently by the non-abelian Čech cohomology

$$\check{\mathrm{H}}^1(X,\underline{\operatorname{Fun}}_{\mathcal{C}}(\mathcal{M},\mathcal{M})^{\mathrm{inv}}).$$

The conjecture is motivated by the hypothesis that the space of systems in the phase is homotopy equivalent to this classifying space; the paper does not give a general proof.