Asymptotic conjecture for Euler characteristics of minimal strata of differentials

Let ${\mathcal{M}}_g(2g-1)$ denote the moduli space of minimal holomorphic strata of differentials in genus $g$, and let $\chi$ denote its orbifold Euler characteristic. Asymptotic Euler-characteristic conjecture. For all $g\geq 1$, $\chi({\mathcal{M}}_g(2g-1))$ is negative. Moreover, there exist positive constants $A$ and $B$ such that

$$A\frac{(2g-1)!}{(2g-1)^4}\leq -\chi({\mathcal{M}}_g(2g-1))\leq B\frac{(2g-1)!}{(2g-1)^3}$$

for all $g$. Numerical experiments suggest this growth rate, which is much higher than that of ${\mathcal{M}}_{g,1}$; the conjecture would also indicate that the cohomology of minimal strata is not spanned by tautological classes, although establishing this would require comparing topological and orbifold Euler characteristics.