Cowan's conjecture on the average Mordell–Weil rank of elliptic curves over the rational function field

Let $\mu$ be the Mahler measure on $\mathbf{Z}[t]$, and for a positive integer $d$ and real number $M$ define

$$P_d(M)=\{p\in\mathbf{Z}[T]\mid \deg(p)\leq d,\ \mu(p)<M\}.$$

For positive integers $m,n$ and real $M$, let

$$S_{m,n}(M)=\left\{E_{A,B}: y^2=x^3+A(t)x+B(t)\ \middle|\ A\in P_m(M^2),\ B\in P_n(M^3),\ 4A(t)^3+27B(t)^2\neq 0\right\}.$$

Cowan's conjecture. For every pair of positive integers $m,n$,

$$\lim_{M\to\infty}\frac{1}{\#S_{m,n}(M)}\sum_{E\in S_{m,n}(M)}\operatorname{rank}E(\mathbf{Q}(t))=0.$$

The conjecture concerns the average Mordell–Weil rank of elliptic curves over $\mathbf{Q}(t)$ ordered by the Mahler measures of the coefficients. The source paper states that it proves this conjecture for elliptic surfaces over $\mathbf{Q}$ and establishes an analogous result over arbitrary number fields.