Asymptotic proportions of non-hyperelliptic isogeny classes and obstruction-rule polynomials

Let $\Pi_3(q)$ be the proportion of isogeny classes of $3$-dimensional abelian varieties over $\mathbb{F}_q$ that do not contain the Jacobian of a genus-$3$ hyperelliptic curve. Let $\chi_3(q)$ be the proportion of degree-$6$ $q$-Weil polynomials for which rule $\textbf{1.N.N.0}$ holds when $q$ is odd and rule $\textbf{0.N.N.0}$ holds when $q$ is even. Asymptotic proportion conjecture. The following limits hold:

$$\lim_{\substack{q\to\infty \\ q\ \mathrm{odd}}}\Pi_3(q)=\lim_{\substack{q\to\infty \\ q\ \mathrm{odd}}}\chi_3(q)=\frac{1}{4},$$ $$\lim_{\substack{q\to\infty \\ q\ \mathrm{even}}}\Pi_3(q)=\lim_{\substack{q\to\infty \\ q\ \mathrm{even}}}\chi_3(q)=\frac{1}{2}.$$

These limits quantify the expected asymptotic proportion of non-hyperelliptic isogeny classes and of Weil polynomials detected by the relevant rule; the conjecture is part of the paper's statistical analysis and remains open.