Asymptotic normality conjecture for unordered rational-curve moduli spaces

Let $\overline{\mathcal M}_{0,n}$ be the moduli space of stable rational curves with $n$ marked points, let $\overline{\mathcal M}_{0,n+1}$ have one distinguished marked point fixed by the action, and let $\mathbb{P}^1[n]$ be the Fulton–MacPherson configuration space. The symmetric group $\mathbb{S}_n$ acts by permuting the marked points, and consider the quotient spaces

$$\overline{\mathcal M}_{0,n}/\mathbb{S}_n,\qquad \overline{\mathcal M}_{0,n+1}/\mathbb{S}_n,\qquad \mathbb{P}^1[n]/\mathbb{S}_n.$$

Asymptotic normality conjecture. The Betti numbers of these three quotient spaces are asymptotically normally distributed, and the associated variances grow linearly at the same rate.

This conjecture extends the established asymptotic normality of the Betti numbers for the corresponding spaces with ordered marked points. Numerical data and asymptotic log-concavity support the claim, but the asserted normality and common linear variance growth remain unproved.