Rigidity conjecture for fusion rings and tensor functors

Let ${\bf k}$ be a field, and let a fusion ring mean the Grothendieck ring of a fusion category. A realization of a fusion ring over ${\bf k}$ is a fusion category over ${\bf k}$ whose Grothendieck ring is the given fusion ring; realizations are considered up to equivalence. For two fixed fusion categories, consider tensor functors between them up to natural tensor isomorphism.

Rigidity conjecture. (i) Any fusion ring has at most finitely many realizations over ${\bf k}$, up to equivalence (possibly none). (ii) The number of tensor functors between two fixed fusion categories, up to a natural tensor isomorphism, is finite.

This conjecture concerns the finiteness of categorical structures realizing fixed fusion data and of tensor functors between fixed fusion categories. It is open in general, but holds for categories over ${\mathbb C}$ and hence for all fields of characteristic zero.