Asymptotic dominance conjecture for Euler characteristics of relative stable-map spaces

Let $\overline{\mathcal{M}}_n$ denote the moduli space considered in the paper, and let $\overline{\mathcal{M}}_{0,n+1}$ denote the corresponding genus-zero moduli space. Their Euler characteristics are written $\chi(\overline{\mathcal{M}}_n)$ and $\chi(\overline{\mathcal{M}}_{0,n+1})$. Asymptotic dominance conjecture. One has

$$\lim_{n \to \infty} \frac{\chi(\overline{\mathcal{M}}_{0,n+1})}{\chi(\overline{\mathcal{M}}_n)} = 0.$$

The conjecture formalizes the observed much faster growth of $\chi(\overline{\mathcal{M}}_n)$ than $\chi(\overline{\mathcal{M}}_{0,n+1})$. The latter has a known asymptotic formula, but an asymptotic analysis of $\chi(\overline{\mathcal{M}}_n)$, and hence a proof of this limit, remains open.