On numbers n with polynomial image coprime with the nth term of a linear recurrence

Let $F$ be an integral linear recurrence, $G$ be an integer-valued polynomial splitting over the rationals, and $h$ be a positive integer. Also, let $mathcal{A}_{F,G,h}$ be the set of all natural numbers $n$ such that $gcd(F(n), G(n)) = h$. We prove that $mathcal{A}_{F,G,h}$ has a natural density. Moreover, assuming $F$ is non-degenerate and $G$ has no fixed divisors, we show that $mathbf{d}(mathcal{A}_{F,G,1}) = 0$ if and only if $mathcal{A}_{F,G,1}$ is finite.