Várilly-Alvarado and Vakil's limiting measure conjecture for scrollar invariants

Fix $d \geq 2$. Let $\mathcal{H}_{d,g}$ be the stack of smooth degree-$d$, genus-$g$ covers of $\mathbb{P}^1$, and let $\mathcal{H}_{\mathbf{e},g}$ be the substack with scrollar invariants $\mathbf{e}$. For an open ball $B \subseteq \mathbb{R}^{d-1}$ and $g>0$, define

$$\pi^{\textnormal{geo}}(B,g) = \frac{1}{\dim \mathcal{H}_{d,g}}\dim\left(\bigcup_{\mathbf{e} \in \mathbb{Z}^{d-1}:\,\mathbf{e}/(d+g-1)\in B}\mathcal{H}_{\mathbf{e},g}\right).$$

Várilly-Alvarado and Vakil's limiting conjecture. For all $B \subseteq \mathbb{R}^{d-1}$, the limit $\lim_{g \to \infty}\pi^{\textnormal{geo}}(B,g)$ exists and depends only on the intersection of $B$ with the hyperplane

$$\left\{(\overline{e}_1,\dots,\overline{e}_{d-1}):\sum\overline{e}_i=1\right\}$$

and not on $B$. This conjecture describes the limiting geometric distribution of scrollar invariants; it is known in the paper for $d=5$, but remains open in general.