Sundaram–Welker vanishing conjecture for polynomial root-multiplicity strata

Let $n\geq 2$, let $\lambda=(\lambda_1,\dots,\lambda_t)\vdash n$ be a number partition, and let $\Sigma_\lambda^{S^2}$ be the one-point compactification of the space of monic degree-$n$ complex polynomials whose roots can be partitioned into sets of sizes $\lambda_1,\dots,\lambda_t$, with equal roots within each set. Write $l(\lambda)=t$ and let $\tilde\beta_i(\Sigma_\lambda^{S^2},{\mathbb Q})$ denote its reduced rational Betti numbers.

Sundaram–Welker's conjecture. For every number partition $\lambda$,

$$\tilde\beta_i(\Sigma_\lambda^{S^2},{\mathbb Q})=0$$

unless $i=2l(\lambda)$.

This predicts concentration of the reduced rational homology of these root-multiplicity strata in a single degree; the supplied text attributes the conjecture to Sundaram and Welker but gives no resolution status.