Non-trivially graded self-dual fusion categories of rank 4

Let C be a self-dual spherical fusion categories of rank 4 with non-trivial grading. We complete the classification of Grothendieck ring K(C) of C; that is, we prove that K(C) ≌ Fib⊗Z[Z₂], where Fib is the Fibonacci fusion ring and Z[Z₂] is the group ring on Z₂. In particular, if C is braided, then it is equivalent to FibVec_Z^ω₂ as fusion categories, where Fib is a Fibonacci category and FibVec_Z^ω₂₂ is a rank 2 pointed fusion category.