Stability criterion for exceptional post-critically finite quadratic polynomials

Let $S$ be the set given in Proposition 3.8. For $m\in S$, define

$$f(x)=(x-m^2)^2+m^2-1\in\mathbb{Z}[x].$$

Stability conjecture. For every $m\in S$, the polynomial $f$ is stable, meaning that every iterate $f^n$ is irreducible over $\mathbb{Q}$. This conjecture concerns the three values excluded from Proposition 3.8; it is based on extensive computations in MAGMA, and no proof is supplied in the source.