The strata-generation conjecture for Hurwitz-space cohomology

Let ${\mathbb P}{\cal H}_n$ be the projectivized Hurwitz space, and let $C_n$ and $M_n$ be the cohomology classes of the caustic and Maxwell stratum, respectively. Consider the subalgebra of $H^*({\mathbb P}{\cal H}_n)$ generated by the cohomology classes of the strata. Strata-generation conjecture. This subalgebra is generated by the two classes $C_n$ and $M_n$ in $H^2({\mathbb P}{\cal H}_n)$. This would reduce the study of the relevant symmetric cohomology to the two distinguished codimension-one strata, enabling the intersection calculations used in the paper; the source gives no resolution.