For a rational function f, and a base point a, both defined over some field K, the arboreal Galois representation associated to the pair (f, a) is the homomorphism describing the action of the absolute Galois group of K on the tree of iterated preimages of a by f. Generically, one expects the image of Galois to be fairly large in the (graph-theoretic) automorphism group of this tree, but there are pathological cases in which is quite a bit smaller. For instance, if f(z)=z^d and a is a root of unity, then the action of Galois is abelian, while the automorphism group of the tree is very far from being so. In 2020, Andrews and Petsche conjectured that, over a number field, the only examples that give rise to abelian arboreal Galois representations are preimages of roots of unity by the power map, and a similar construction involving Chebyshev polynomials. This talk will present some results in this direction in the function field case, obtained in joint work with Pagano and Ferraguti, as well as some new exceptional cases in the conjecture in this context.