Layered degree-one symmetric homology conjecture for the polynomial algebra

Let $k$ be a commutative ring, let $k[t]$ be the polynomial algebra, and for $m\geq 0$ let $HS_1(k[t])_{t^m}$ denote its $t^m$-layered symmetric homology, obtained from the decomposition by products of tensor factors. Polynomial-layer conjecture.

$$HS_1\left(k[t]\right)_{t^m}=\begin{cases}0,&m=0,1,\\ k/2k,&m\geq 2. \end{cases}$$

The conjecture is based on computations using cyclic monoids and describes the degree-one contribution of every monomial layer, but no proof for all $m$ is supplied.