Heuristic for counting principally polarized abelian surfaces over the rationals

Let $\mathcal{N}_{2,\mathbb{Q}}(\mathcal{B})$ count principally polarized abelian surfaces $A=\operatorname{Jac}(X)$, where $X$ is a stable genus $2$ curve over $\mathbb{Z}$ satisfying $0<\operatorname{ht}(\Delta_2(X))\leq\mathcal{B}$. Heuristic for $\mathcal{N}_{2,\mathbb{Q}}(\mathcal{B})$. There are constants $a,b,c$ such that

$$\mathcal{N}_{2,\mathbb{Q}}(\mathcal{B})\sim_{\mathcal{O}(1)}a\mathcal{B}^{\frac{7}{10}}+b\mathcal{B}^{\frac{3}{10}}+c.$$

The claim is motivated by the global-fields analogy and the corresponding finite-field counting results; it is presented as a heuristic, and no resolution is supplied in the source.