Quadratic polynomial primitive-divisor density conjecture

Let $\phi\in\mathbb{Z}[x]$ be quadratic, with infinite critical orbit and all iterates irreducible. For an integer sequence $a_n=\phi(a_{n-1})$, let $P(a_n)$ denote the largest prime factor of $a_n$, and let $D$ denote the associated natural density. Quadratic polynomial primitive-divisor density conjecture. For every $a_0\in\mathbb{Z}$,

$$D(P(a_n))=0.$$

The conjecture is motivated by computations and heuristics and is also identified as Conjecture 5.7 of [13]; no resolution is given in the source.