Ault–Fiedorowicz conjecture on the topological formula for symmetric homology

Let $k$ be a commutative coefficient ring, let $A$ be a $k$-algebra, and let $\mathrm{HS}_*(A)$ denote its symmetric homology. Let $D$ be the monad of an $E_{\infty}$-operad in simplicial $k$-modules corresponding to the classical Barratt–Eccles operad, let $T$ be the tensor algebra functor on $k$-modules, and let $B(D,T,A)$ denote the associated two-sided bar construction.

Ault–Fiedorowicz conjecture. For every $k$-algebra $A$, there is an isomorphism

$$\mathrm{HS}_*(A)\cong \mathrm{H}_*(B(D,T,A),k).$$

This conjecture gives a proposed topological formula for the symmetric homology of an arbitrary algebra. The source says that it apparently remains unproven, while noting that it is proved when $k$ is a field of characteristic $0$; a cited later remark also suspects that the general statement may be false.