Ault–Fiedorowicz conjecture on the symmetric homology of polynomial algebras

Let $k$ be the coefficient ring, let $n\geq 1$, and let $\mathrm{HS}_*(A)$ denote the symmetric homology of a $k$-algebra $A$. Let $\Omega^{\infty}\Sigma^{\infty}$ be the stable homotopy functor, and let $\mathcal{C}_{\infty}$ be the monad associated to the little $\infty$-cubes operad, both acting on based topological spaces $\mathrm{Top}_*$. Write $\mathbb{S}^0$ for the zero-dimensional sphere.

Ault–Fiedorowicz conjecture. For every $n\geq 1$, there is an isomorphism of graded algebras

$$\mathrm{HS}_*(k[x_1,\ldots,x_n])\cong \mathrm{H}_*\left(\prod_{i=1}^n\mathcal{C}_{\infty}(\mathbb{S}^0)\times\prod_{i=2}^n\Omega^{\infty}\Sigma^{\infty}(\mathbb{S}^{i-1})^{\binom{n}{i}},k\right).$$

The conjecture proposes a topological description of symmetric homology for polynomial algebras, whose computation is otherwise difficult even for polynomial rings in at least two variables. The source reports that the conjecture remains open.