Equivalence conjecture for algebraic and geometric Chern–Simons constructions

Let $M$ be a connected, oriented, closed Riemannian $n$-manifold with $\HDR^1(M)=0$. Let $\DR(M)$ be its de Rham complex, $\HDR(M)$ its de Rham cohomology, and $\VansQuotient(\VansSmall(\DR(M)) )$ the non-degenerate quotient of the small subalgebra. Let $\PMC$ be the Chern–Simons Maurer–Cartan element associated with an admissible Hodge propagator $\Prpg$, and let $\MC$ denote the canonical Maurer–Cartan element. Equivalence conjecture. There is an admissible Hodge propagator $\Prpg$ such that the $\IBLInfty$-algebras

$$\dIBL^\PMC(\CycC(\HDR(M)))\quad\text{and}\quad\dIBL^\MC(\CycC(\VansQuotient(\VansSmall(\DR(M)))))$$

are $\IBLInfty$-homotopy equivalent. This is intended to identify the geometric perturbative Chern–Simons model with the algebraic canonical model; the source describes the comparison as conjectural.