Francis–Gaitsgory's Koszul duality conjecture

Let $\mathcal{O}$ be an operad in an ambient $\infty$-category $\mathcal{C}$, and let $B\mathcal{O}$ denote its bar construction, a cooperad. Write $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})$ for the $\infty$-category of $\mathcal{O}$-algebras, and let $\mathrm{coAlg}^{\mathrm{dp},\mathrm{nil}}_{B\mathcal{O}}(\mathcal{C})$ be the $\infty$-category of conilpotent $B\mathcal{O}$-coalgebras with divided powers. An $\mathcal{O}$-algebra is pronilpotent if it belongs to the smallest class of $\mathcal{O}$-algebras closed under limits and containing all nilpotent $\mathcal{O}$-algebras. The functors $\mathrm{indec}_{\mathcal{O}}^{\mathrm{nil}}$ and $\mathrm{prim}^{\mathrm{nil}}_{B\mathcal{O}}$ are adjoint, where the latter is the right adjoint of the former. Francis–Gaitsgory's conjecture. The adjoint functors $\mathrm{indec}_{\mathcal{O}}^{\mathrm{nil}}$ and $\mathrm{prim}^{\mathrm{nil}}_{B\mathcal{O}}$ restrict to an adjoint equivalence from the full subcategory of $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})$ on pronilpotent algebras to $\mathrm{coAlg}^{\mathrm{dp},\mathrm{nil}}_{B\mathcal{O}}(\mathcal{C})$. This is a precise form of operadic Koszul duality, extending the familiar bar–cobar duality between algebras and coalgebras. The source presents it as a conjecture of Francis and Gaitsgory; no resolution status is supplied here.