Cohen's Gerstenhaber structure conjecture for double loop spaces

Let $X$ be a $2$-connected pointed topological space, let $\Omega_M X$ denote its pointed Moore loop space, and let $S_*(\Omega_M X)$ be its singular-chain differential graded bialgebra. The groups $\operatorname{Cotor}_{S_*(\Omega_M X)}(\Bbbk,\Bbbk)$ and $H_*(\Omega^2X)$ carry the Gerstenhaber algebra structures specified in the source.

Cohen's Gerstenhaber structure conjecture. There is an isomorphism of Gerstenhaber algebras

$$\operatorname{Cotor}_{S_*(\Omega_M X)}(\Bbbk,\Bbbk)\cong H_*(\Omega^2X).$$

Here the structure on the left is the one supplied by the cited theorem, while the structure on the right is the usual one given by Cohen.

The conjecture asserts that the Gerstenhaber structure obtained through the cobar construction agrees with the standard double-loop-space structure. The source presents this as an expected identification rather than a proved result.