The polynomiality conjecture for strata classes in Hurwitz spaces

Let ${\mathbb P}{\cal H}_n$ be the projectivized Hurwitz space. Let $\Psi_n,\Delta_n\in H^2({\mathbb P}{\cal H}_n)$, where $\Psi_n$ is the distinguished class used for intersection computations and $\Delta_n$ is dual to the subvariety of functions defined on singular curves. Polynomiality conjecture. Each cohomology class dual to a stratum in ${\mathbb P}{\cal H}_n$ can be expressed as a homogeneous polynomial in $\Psi_n$ and $\Delta_n$ whose coefficients are polynomials in $n$. This proposed sharpening of Kazarian's principle would make all stratum intersection classes uniform in $n$; the source gives no resolution.