Stable first-homology representation conjecture for complete graph configuration spaces

Let $K_n$ be the complete graph on $n$ vertices, let $\operatorname{Conf}_k(K_n)$ denote the ordered configuration space of $k$ particles on $K_n$, and let $S_n$ act by permuting the vertices. Write $\mathrm{Sp}_n(1,1)$ for the irreducible $S_n$-representation indexed by the partition $(n-2,1,1)$.

Stable first-homology representation conjecture. For $n\geq 5$, there is an isomorphism of $S_n$-representations

$$H_1(\operatorname{Conf}_k(K_n))\cong k\,\mathrm{Sp}_n(1,1).$$

The conjecture is motivated by computed homology representations for configuration spaces of one, two, and three particles on complete graphs. It predicts a uniform description of first homology for every particle number $k$, beyond the cases accessible in the reported computations; no resolution is supplied in the source.