Stability criterion for post-critically finite quadratic polynomials

Let $K$ be a field of characteristic different from $2$, and let $f(x)\in K[x]$ be a monic, post-critically finite quadratic polynomial. Let $o_f$ denote the period of the post-critical orbit and $t_f$ its tail length. A polynomial is stable when every iterate $f^n$ is irreducible over $K$. Stability criterion. The following equivalences should hold:

1. If $t_f=0$, then $f$ is stable if and only if $f^{o_f+1}$ is irreducible.
2. If $t_f>0$, then $f$ is stable if and only if $f^{o_f}$ is irreducible.

This conjecture proposes a necessary and sufficient irreducibility test for stability over arbitrary fields of characteristic not equal to $2$. The source presents it after proving the case $t_f=1$; the general cases remain open there.