Average rank zero for elliptic surfaces ordered by coefficient size

For positive integers $d$ and $M$, let

$$\mathcal{P}_d(M)=\{p\in\mathbb{Z}[T]:\deg(p)=d,\ \mu(p)<M\},$$

where $\mu(p)$ is the Mahler measure, and define

$$\mathcal{S}_{m,n}(M)=\{\mathcal{E}:y^2=x^3+A(T)x+B(T):A\in\mathcal{P}_m(M^2),\ B\in\mathcal{P}_n(M^3),\ 4A(T)^3+27B(T)^2\not\equiv0\}.$$

Here $\mathcal{S}_{m,n}=\mathcal{S}_{m,n}(M)$ is the family of elliptic surfaces with fixed positive degrees $m,n$. Average-rank-zero conjecture. For any fixed positive integers $m$ and $n$,

$$\lim_{M\to\infty}\frac{1}{\#\mathcal{S}_{m,n}(M)}\sum_{\mathcal{E}\in\mathcal{S}_{m,n}(M)}\operatorname{rank}\mathcal{E}(\mathbb{Q}(T))=0.$$

This is the paper's precise formulation of the claim that $100\%$ of elliptic surfaces have rank zero under ordering by the size of the Weierstrass coefficients. No resolution evidence is supplied in the source.