Intersection of orbits for polynomials in characteristic p

In [GTZ08, GTZ12], the following result was established: given polynomials f,g[x] of degrees larger than $1$, if there exist α,β such that their corresponding orbits Of(α) and Og(β) (under the action of f, respectively of g) intersect in infinitely many points, then f and g must share a common iterate, i.e., fᵐ=gⁿ for some m,n. If one replaces C with a field K of characteristic p, then the conclusion fails; we provide numerous examples showing the complexity of the problem over a field of positive characteristic. We advance a modified conjecture regarding polynomials f and g which admit two orbits with infinite intersection over a field of characteristic p. Then we present various partial results, along with connections with another deep conjecture in the area, the dynamical Mordell-Lang conjecture.