Fully faithful functor conjecture for Hochschild cohomology of fully dualizable objects

Let $\mathcal{C}$ be a fully dualizable object, and let $Z(\mathcal{C})=HH^\bullet(\mathcal{C})$ denote its Hochschild cohomology. Consider a category of fully dualizable objects and a target category constructed using a system of cospans. Fully faithful Hochschild-cohomology conjecture. The assignment $Z(-)$ defines a functor from the category of fully dualizable objects to this target category, and it is fully faithful after restricting to $0$-morphisms and equivalence classes of invertible $1$-morphisms in the domain while ignoring higher morphisms. In particular, for fully dualizable objects $\mathcal{C}$ and $\mathcal{D}$, $Z(\mathcal{C})$ is isomorphic to $Z(\mathcal{D})$ as, for example, $E_n$-categories when appropriate, if and only if $\mathcal{C}$ and $\mathcal{D}$ are Morita equivalent; moreover, there is a group isomorphism

$$\operatorname{Pic}(\mathcal{C})\simeq\operatorname{Aut}(Z(\mathcal{C})).$$

This proposes an extension of the analogous properties observed for Levin–Wen models and rational conformal field theories to general extended topological quantum field theories. The claim is presented as a mathematical conjecture, and no resolution is supplied in the source.