I am not an expert, but what I think I know on the subject is:
For $GL_n$, the local Langlands conjecture has long been known by work of Laumon, Rapoport and Stuhler (Inventiones Math, 1993). Of course, in this case, even the global correspondence is now known, due to the Fields-Medal-winning work of Laurent Lafforgue.
For general reductive groups, the local Langlands correspondence is not known
at this date but there is movement right now.
Vincent Lafforgue (Laurent's younger brother) has recently released a paper
proving the direction "automorphic --> Galois" of the global correspondence. In this paper, he announces a work in preparation of himself and Genestier
aiming at establishing the local Langlands correspondence for reductive groups.
So when this paper is released, the answer to your question may well be "solved!".