Heuristic for counting elliptic curves with prescribed level structures

Let $\mathcal{Z}_{1,\mathbb{Q}}^{[\Gamma]}(\mathcal{B})$ count generalized elliptic curves with $[\Gamma]$-structures over $\mathbb{Z}$ ordered by $0<\operatorname{ht}(\Delta_1)\leq\mathcal{B}$. Heuristic for elliptic curves with level structures. The following asymptotic-tightness estimates are expected:

$$\begin{aligned} \mathcal{Z}_{1,\mathbb{Q}}^{[\Gamma_1(2)]}(\mathcal{B})&\sim_{\mathcal{O}(1)}a\mathcal{B}^{\frac12}+b,\\ \mathcal{Z}_{1,\mathbb{Q}}^{[\Gamma_1(3)]}(\mathcal{B})&\sim_{\mathcal{O}(1)}a\mathcal{B}^{\frac13}+b,\\ \mathcal{Z}_{1,\mathbb{Q}}^{[\Gamma_1(4)]}(\mathcal{B})&\sim_{\mathcal{O}(1)}a\mathcal{B}^{\frac14}+b,\\ \mathcal{Z}_{1,\mathbb{Q}}^{[\Gamma(2)]}(\mathcal{B})&\sim_{\mathcal{O}(1)}a\mathcal{B}^{\frac13}+b. \end{aligned}$$

These estimates arise in the appendix from finite-field counting results and the global-fields analogy; the source gives no resolution.