The equivariant infinity-category construction conjecture

Let $G$ be a smooth algebraic group and let $X$ be a left $G$-variety over $\operatorname{Spec} K$. Let $\mathsf{Ner}(\operatorname{\mathbf{SfResl}}_G(X)^{\operatorname{op}})$ be the simplicial nerve of $\operatorname{\mathbf{SfResl}}_G(X)^{\operatorname{op}}$, and let $F$ be a pre-equivariant $\infty$-pseudofunctor on $X$, namely an $\infty$-functor

$$F:\mathsf{Ner}(\operatorname{\mathbf{SfResl}}_G(X)^{\operatorname{op}})\to\mathsf{Ner}_{\operatorname{hc}}(\mathfrak{QCat}).$$

For $\Gamma\in\operatorname{\mathbf{Sf}}(G)_0$, write $\,_{\Gamma}{\mathnormal{X}}$ for the corresponding variety, and for $f\in\operatorname{\mathbf{Sf}}(G)_1$ write $\overline{\mathnormal{f}}$ for its induced map. The equivariant infinity-category construction conjecture. To $F$ there is an associated equivariant $\infty$-category $F_G(X)$ whose $0$-cells are pairs $(A,T_A)$, where

$$A=\lbrace\,_{\Gamma}{\mathnormal{A}}\,\rbrace_{\Gamma\in\operatorname{\mathbf{Sf}}(G)_0}$$

with each $\,_{\Gamma}{\mathnormal{A}}$ a $0$-cell of $F(\,_{\Gamma}{\mathnormal{X}})$, and

$$T_A=\lbrace\tau_f^A:F(\overline{\mathnormal{f}})(\,_{\Gamma^{\prime}}{\mathnormal{A}})\xrightarrow{\simeq}\,_{\Gamma}{\mathnormal{A}}\rbrace_{f\in\operatorname{\mathbf{Sf}}(G)_1},$$

subject to a higher cocycle condition involving fillers of the desired diagrams. This would provide the missing rigorous construction of the equivariant $\infty$-category, from which equivariant derived $\infty$-categories and equivariant $\infty$-categories of perverse sheaves can be obtained. The supplied text gives no evidence that the conjecture has been resolved.