  We show that the Vojta (or Hall-Lang) conjecture implies that the arboreal
Galois representations in a 1-parameter family of quadratic polynomials are
surjective if and only if they surject onto some finite and uniform quotient.
As an application, we use the Vojta conjecture, our uniformity theorem over
$\mathbb{Q}(t)$, and Hilbert's irreducibility theorem to prove that the prime
divisors of many quadratic orbits have density zero.
