The asymptotic inequality conjecture for Bézout coefficients and resultants

For positive integers $m,n,H$, let

$$S_{m,n}(H)=\{(f,g)\in\mathbb{Z}[x]^2:\deg(f)=m,\ \deg(g)=n,\ L(f)>0,\ L(g)>0,\ f\text{ and }g\text{ are coprime},\ \mathrm{H}(f)\leq H,\ \mathrm{H}(g)\leq H\},$$

where $\mathrm{H}(h)$ is the maximum absolute value of the coefficients of $h$ and $L(h)$ is its leading coefficient. For coprime integer polynomials $f,g$, let $B(f,g)$ and $R(f,g)$ denote the integers defined in the paper. Asymptotic inequality conjecture. For any positive integers $m$ and $n$ with $m\geq2$ and $n\geq2$, we have

$$\lim_{H \to \infty}\frac{|\{(f,g)\in S_{m,n}(H): \, B(f,g) = R(f,g)\}|}{|S_{m,n}(H)|} < \frac{1}{2}.$$

The conjecture records the numerical observation that, for polynomial degrees at least two, equality between $B(f,g)$ and the resultant-related integer $R(f,g)$ should occur for fewer than half of randomly chosen coprime pairs; the paper gives computational evidence but no proof.