Odoni–Stoll conjecture on stable reduction of x^2+1

Let $\overline{f}$ denote the reduction of a polynomial $f$ modulo a prime. Odoni–Stoll conjecture. For

$$F=\mathbb{Q},\qquad f(x)=x^2+1,$$

$\overline{f}$ is stable for $p=3$ and for no other primes. Odoni observed stability at $p=3$, while Stoll proved that the associated arboreal representation is surjective; the stated assertion concerns the complete list of stable primes and is presented in the source as a conjecture.