Suppose that C is a ribbon monoidal category with dominant ribbon functors F_1: C->D_1 and F_2: C->D_2 such that D_1 and D_2 are modular tensor categories, does it follow that D_1 and D_2 are equivalent as MTCs? Here dominant means that every object in the target is a summand of an object in the image of the functor.
This is certainly true if C is premodular (semisimple with finitely many simple objects) as was proved by Bruguieres. What if C is not premodular? I haven't been able to locate a more general statement in the literature.
The particular case I have in mind is where C is the Kuperberg G_2-spider specialized to q a particular root of unity. After semisimplification C is in fact premodular, but actually proving that is likely to be a lot of work (it would require writing down inductive formulas for simples, etc.).