Lekili–Evans comparison conjecture for Hamiltonian action maps

Let $G$ act Hamiltonianly on a monotone symplectic manifold $X$. The action gives an $E_2$-algebra map

$$C_*(\Omega G)\longrightarrow \operatorname{Hoch}^*(\operatorname{Fukaya}(X)).$$

Lekili and Evans' construction gives a map from the endomorphisms of the cotangent fiber in the wrapped Fukaya category of $T^*G$ to the endomorphisms of the diagonal in the Fukaya category of $X^-\times X$; after identifying the relevant codomains, Lekili–Evans comparison conjecture. As $A_\infty$-algebra maps, the map constructed from the Hamiltonian action is homotopic to the map constructed by Lekili and Evans. Part of the conjecture is that this equivalence is intertwined by a natural chain-level, $A_\infty$-algebra equivalence between Hochschild cochains and endomorphisms of the diagonal. The conjecture compares a map built from bundles and localization with one built from Lagrangian correspondences and quilts. The required chain-level identification of codomains is part of the assertion, and no resolution is stated.