Spin Euler-characteristic ratio conjecture for minimal strata

Let $\mathcal{M}_g(2g-1)$ be the minimal stratum of differentials, let $\chi(\mathcal{M}_g(2g-1))^{\rm spin}$ denote its spin Euler characteristic, and let $\chi(\mathcal{M}_g(2g-1)^{\rm even})$ and $\chi(\mathcal{M}_g(2g-1)^{\rm odd})$ denote the Euler characteristics of its even and odd components. Spin asymptotic conjecture. There exist positive constants $A$ and $B$ such that

$$A\frac{g}{2^g}<\frac{\chi(\mathcal{M}_g(2g-1))^{\rm spin}}{-\chi(\mathcal{M}_{g}(2g-1))}<B\frac{g^2}{2^g}$$

for all $g$. In particular, $\chi(\mathcal{M}_g(2g-1))^{\rm spin}>0$ and

$$\frac{\chi(\mathcal{M}_g(2g-1)^{\rm even})}{\chi(\mathcal{M}_g(2g-1)^{\rm odd})}\longrightarrow 1\quad\text{for }g\longrightarrow\infty.$$

This stronger conjecture implies the preceding asymptotic equivalence of the Euler characteristics of the odd and even components and concerns the large-genus behavior of spin strata.