Visibility Density Conjecture for polynomial lines of sight

Let $F(x)\in \mathbb{Z}[x]$ be a polynomial with positive leading coefficient. A lattice point $(a,h)\in\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$ is visible along $F(x)$ if there exists $t\in\mathbb{Q}$ such that $h=tF(a)$ and $a$ is the smallest positive integer $u$ for which $tF(u)$ is a positive integer. Define the density of visible lattice points, when the limit exists, by

$$D(F)=\lim_{N\to\infty}\frac{\#\{(a,b)\in\mathbb{Z}_{>0}^2:\max\{a,b\}\leq N\text{ and }(a,b)\text{ is visible along }F(x)\}}{N^2}.$$

Visibility Density Conjecture. If $F(x)\in\mathbb{Z}[x]$ has at least two distinct roots, then the set of lattice points in $\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$ visible along $F(x)$ has density $1$; equivalently,

$$D(F)=1.$$

This extends the conjecture of Chaubey and Pandey for polynomials passing through the origin. It predicts a sharp contrast with monomials, for which the visible-point density is less than $1$, and asserts that almost all lattice points are visible when the polynomial has at least two distinct roots.