Half positive-rank density for elliptic surfaces over finite fields

For positive integers $m$ and $n$, let $\mathcal{S}_{\ell;m,n}$ be the finite set of elliptic surfaces over $\mathbb{F}_\ell$ of the form $y^2=x^3+A(T)x+B(T)$ with $\deg(A)=m$ and $\deg(B)=n$. Let $\rho_\ell(m,n)$ be the proportion of surfaces in this set having positive rank. Half-density conjecture. For every prime $\ell_0$ and every pair of integers $m_0,n_0>0$,

$$\lim_{\ell\to\infty}\rho_\ell(m_0,n_0)=\lim_{m\to\infty}\rho_{\ell_0}(m,n_0)=\lim_{n\to\infty}\rho_{\ell_0}(m_0,n)=\frac12.$$

This conjecture predicts that positive rank has limiting density one half in each of three asymptotic regimes. The source gives no evidence of resolution.