Formality conjecture for the geometric Chern–Simons construction

Let $M$ be a connected, oriented, closed Riemannian manifold with $\HDR^1(M)=0$, and suppose that $M$ is formal. In the situation of the equivalence conjecture, let $\PMC$ be the Chern–Simons Maurer–Cartan element. Geometric formality conjecture. The twisted algebra $\dIBL^\PMC(\CycC(\HDR(M)))$ is $\IBLInfty$-homotopy equivalent to $\dIBL^\MC(\CycC(\HDR(M)))$. This is presented as a potentially easier consequence of the algebraic–geometric equivalence conjecture; no proof is supplied.