The Schwartz genus conjecture for polynomial equations

Let $q_n$ denote the polynomial equation problem associated with the unordered configuration space of $n$ points, let $g(q_n)$ be its Schwartz genus, and let $L_n$ be the relevant coefficient system for the symmetric group $\operatorname{\Sigma}_n$. The conjecture concerns those $n$ for which the group

$$\operatorname{H}_{n-1}(\Sigma_n;L_n)$$

is non-trivial. Schwartz genus conjecture. For every such $n$,

$$g(q_n)=n.$$

The paper explains that this is equivalent to the corresponding class in the Serre spectral sequence surviving when the indicated homology group is non-zero. The question is presented as open and concerns the Schwartz genus of the polynomial-equation problem.