Heuristic for counting hyperelliptic curves over the rationals

For $g\geq3$, let $\mathcal{Z}_{g,\mathbb{Q}}(\mathcal{B})$ count stable hyperelliptic genus $g$ curves with a marked rational Weierstrass point over $\mathbb{Z}$ satisfying $0<\operatorname{ht}(\Delta_g)\leq\mathcal{B}$. Heuristic for $\mathcal{Z}_{g,\mathbb{Q}}(\mathcal{B})$. There are constants $a,b,c,d,e$ such that

$$\mathcal{Z}_{3,\mathbb{Q}}(\mathcal{B})\sim_{\mathcal{O}(1)}a\mathcal{B}^{\frac{9}{14}}+b\mathcal{B}^{\frac{2}{7}}+c\mathcal{B}^{\frac{3}{14}}+d,$$ $$\mathcal{Z}_{4,\mathbb{Q}}(\mathcal{B})\sim_{\mathcal{O}(1)}a\mathcal{B}^{\frac{11}{18}}+b\mathcal{B}^{\frac{5}{18}}+c\mathcal{B}^{\frac{1}{4}}+d\mathcal{B}^{\frac{1}{6}}+e.$$

For $g\geq5$, the corresponding degrees can similarly be worked out through the cited theorem. This is a global-fields heuristic based on finite-field estimates; the source does not provide a resolution.