The frequency conjecture for Bézout coefficients of random integer polynomials

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 of the absolute values 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. Frequency conjecture. For any positive integers $m$ and $n$, we have

$$\lim_{H \to \infty}\frac{|\{(f,g)\in S_{m,n}(H): \, B(f,g) = r(f,g)\}|}{|S_{m,n}(H)|} \geq \frac{9}{10}.$$

The conjecture formalizes the observed frequency with which the Bézout-related integers $B(f,g)$ and $r(f,g)$ coincide for randomly chosen coprime integer polynomials; the numerical data in the paper suggest frequencies above $90\%$, while no proof of the asserted lower bound is given.