Quantitative rank distribution conjecture for elliptic curves over $\mathbb{F}_q(t)$

Let $n\in\mathbb{Z}_{\geq 0}$ and suppose that $\operatorname{char}(\mathbb{F}_q)>3$. Let $\mathcal{N}^{r}_{T}(\mathbb{F}_q(t),B)$ count the number of $\mathbb{F}_q$-isomorphism classes of minimal elliptic curves over $\mathbb{P}^1_{\mathbb{F}_q}$ with algebraic rank $r\in\mathbb{Z}_{\geq 0}$ and torsion subgroup $T$, ordered by the multiplicative discriminant height $ht(\Delta)=q^{12n}\leq B$. Quantitative rank distribution conjecture.

$$\mathcal{N}^{r=0}_{T=0}(\mathbb{F}_q(t),B)=\left(\frac{q^{9}-1}{q^{8}-q^{7}}\right)B^{5/6}+o(B^{5/6}),$$ $$\mathcal{N}^{r=1}_{T=0}(\mathbb{F}_q(t),B)=\left(\frac{q^{9}-1}{q^{8}-q^{7}}\right)B^{5/6}+o(B^{5/6}),$$ $$\mathcal{N}^{r\geq 2}_{T}(\mathbb{F}_q(t),B)=o(B^{5/6}),$$

where all $o$ terms are little-o. In particular, $|E(K)|=1$ and $E(K)=\mathbb{Z}$ each correspond to $50\%$ of all elliptic curves over $K=\mathbb{F}_q(t)$ ordered by discriminant height, with equal leading term and identical leading coefficient $\left(\frac{q^{9}-1}{q^{8}-q^{7}}\right)$. The exact counting functions for ranks $0$ and $1$ nevertheless differ because they have distinct lower-order main terms. The conjecture is presented as the quantitative consequence over $\mathbb{F}_q(t)$ of the preceding rank-distribution conjecture together with the exact counting and torsion-density results; the source gives no resolution.