Conjecture on the symmetric homology of polynomial rings

Let $k[t_1,t_2,\dots,t_n]$ be the polynomial ring in $n$ indeterminates over $k$, and let $C_\infty$ denote the monad associated to an $E_\infty$ operad. Write $S^i$ for the $i$-sphere and $\Omega^\infty S^\infty(S^{i-1})$ for the corresponding infinite loop space. Symmetric homology conjecture. The symmetric homology of the polynomial ring is

$$HS_*(k[t_1,t_2,\dots,t_n])=H_*\left(\prod_{i=1}^n C_\infty(S^0)\times\prod_{i=2}^n\prod_{j=1}^{n!/(i!(n-i)!)}\Omega^\infty S^\infty(S^{i-1});k\right).$$

This conjecture is motivated by the preceding computation for Laurent polynomial rings and the relation between symmetric homology and group completion for commutative monoids; the supplied text does not indicate whether it has been resolved.