Heuristic conjecture for the limiting distribution of a-numbers of hyperelliptic curves in characteristic three

Continue with $p=3$, let $F={{\mathbb{F}_q}}(X)$, and for $\epsilon\in\{1,2\}$ let $\mu'_{\epsilon,g}(a)$ denote the proportion of genus-$g$ hyperelliptic curves in the corresponding parity class whose $a$-number is $a$, where $a\in\mathbb{Z}_{\geq 0}$. Heuristic conjecture. For $\epsilon\in\{1,2\}$ and $a\in\mathbb{Z}_{\geq0}$,

$$\lim_{g\to\infty}\mu'_{\epsilon,g}(a)=\begin{cases}1-q^{-1}&\text{if }a=0,\q^{-2a+1}(1-q^{-2})&\text{if }a>0. \end{cases}$$

This conjecture proposes that the asymptotic $a$-number distribution agrees with the distribution obtained from random two-dimensional subspaces of fixed height. The preceding theorem establishes the corresponding distribution for the heuristic model, while the asserted connection with hyperelliptic curves remains unproved.