Visibility density conjecture for polynomially defined lattice points

Let $V(a_n,a_{n-1},\dots,a_1)$ denote the set of lattice points associated with the coefficient vector $(a_n,a_{n-1},\dots,a_1)$, and let $\operatorname{dens}$ denote its density. Visibility density conjecture. For every coefficient vector other than $(1,0,\dots,0)$,

$$\operatorname{dens}(V(a_n,a_{n-1},\dots,a_1))=1.$$

The preceding results provide lower bounds for these densities and show that the exceptional vector $(1,0,\dots,0)$ has density $1/\zeta(n+1)$. Numerical experiments motivate the conjecture that all other coefficient vectors have full density.