  Jones conjectures the arboreal representation of a degree two rational map
will have finite index in the full automorphism group of a binary rooted tree
except under certain conditions. We prove a version of Jones' Conjecture for
quadratic and cubic polynomials assuming the $abc$-Conjecture and Vojta's
Conjecture. We also exhibit a family of degree $2$ rational maps and give
examples of degree $3$ polynomial maps whose arboreal representations have
finite index in the appropriate group of tree automorphisms.
