Density-zero conjecture for prime divisors in the orbit of zero under $x^2-1$

Let $f(x)=x^2-1$, and for an integer $n\geqslant 2$ let $P(n)$ denote the set of prime numbers arising in the associated sequence defined in the paper. Density-zero conjecture. For every $n\in {\mathbb Z}_{\geqslant 2}$, $P(n)$ is a set of prime numbers of density zero. The set $P(n)$ is known to be infinite and not to have full density; the conjectured density-zero statement remains open.