The twisting classification conjecture for three-functor formalisms

Let $\mathcal{C}$ be an $\infty$-category, and let

$$\mathcal{D}:\operatorname{Corr}(\mathcal{C})^\otimes\to\operatorname{Cat}_\infty$$

be a three-functor formalism. Let $\mathfrak{LZ}_\mathcal{D}^\otimes$ be its symmetric monoidal $(\infty,2)$-category of cohomological correspondences, with underlying symmetric monoidal $\infty$-category $\operatorname{LZ}_\mathcal{D}^\otimes$. Write $T_\mathcal{C}$ for the twist object and $\mathcal{D}^*$ for the associated functor.

Twisting classification conjecture. There is a canonical isomorphism of anima

$$\operatorname{Hom}_{\operatorname{CAlg}(\operatorname{Cat}_\infty)_{\mathcal{C}^{\operatorname{op},\sqcup}/}}(\operatorname{Corr}(\mathcal{C})^\otimes,\operatorname{LZ}_\mathcal{D}^\otimes)\simeq \operatorname{Hom}_{\operatorname{Fun}(\mathcal{C}^{\operatorname{op}},\operatorname{CAlg}(\operatorname{Cat}_\infty))}(T_\mathcal{C},\mathcal{D}^*).$$

This conjecture aims to characterize the six-functor formalisms obtained by twisting a given formalism. It is presented as an outlook for cases not covered by Scholze's conjecture, and in particular asks for a construction of the twisted three-functor formalisms.