Strong Dynamical Wieferich Prime Conjecture

Let $b\in\frac{1}{2}\mathbb{Z}$ and let $\phi\in\mathbb{Z}[x]$ be separable and quadratic. Assume that the forward orbit $\{\phi^n(b):n=1,2,\ldots\}$ is infinite. Strong Dynamical Wieferich Prime Conjecture. For all but finitely many $n$, there is a prime $p$ such that

$$v_p(\phi^n(b))\text{ is odd},\qquad v_p(\phi^m(b))=0\quad\text{for all }m<n.$$

This primitive-prime-divisor assertion is proposed in connection with finite-index questions for arboreal Galois groups and is attributed in the source to [4]; no resolution is given.