Equivariant monoid realization conjecture for G-connected based G-spaces

Let $G$ be a discrete group, and let $X$ be a based $G$-space. Call $X$ $G$-connected if it satisfies the equivariant connectedness condition used in the source. A discrete $G$-monoid is a monoid with a $G$-action by monoid automorphisms.

Equivariant monoid realization conjecture. Any $G$-connected based $G$-space $X$ is weakly equivalent to

$$B M(X)$$

for some discrete $G$-monoid $M(X)$.

This is presented as a plausible equivariant generalization of results of McDuff and Fiedorowicz; the source notes that Sunny Zhang subsequently proved it. The supplied parser status is unknown, so the database status is left open.