Degree-one symmetric homology conjecture for truncated polynomial algebras

Let $k$ be a commutative ground ring and let $n$ be a nonnegative integer. Consider the truncated polynomial algebra $k[t]/(t^n)$ and its degree-one symmetric homology $HS_1\big(k[t]/(t^n)\big)$. Degree-one truncated-polynomial conjecture.

$$HS_1\big(k[t]/(t^n)\big) = \begin{cases} (k/2k)^n, & \text{if } n\geq 0 \text{ is even},\\ (k/2k)^{n-1}, & \text{if } n\geq 1 \text{ is odd}. \end{cases}$$

The formula is motivated by explicit computations over the integers for $n\leq 6$. Its validity for general commutative ground rings and all indicated values of $n$ remains open.