Boston–Jones model conjecture outside exceptional orbit types

Let $f e \mathbb{F}_q$ be a monic quadratic polynomial, where $\mathbb{F}_q$ is a finite field of odd characteristic. Say that $f$ has orbit type $(3,1)$ or $(2,n)$ when its post-critical orbit has the corresponding orbit type.

Boston–Jones exceptional-orbit conjecture. The distribution of the factorization process of $f$ converges to limiting distributions arising from the Boston–Jones model unless $f$ has orbit type $(3,1)$ or $(2,n)$ for some $n e 1$.

The conjecture is presented as an explanation for the exceptional families observed empirically; the supplied text does not establish its general resolution.