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..[.2], where Fib is the Fibonacci fusion ring and .[.2] is the group ring on .2. In particular, if C is braided, then it is equivalent to Fib.Vec.Z2 as fusion categories, where Fib is a Fibonacci category and Vec.Z2 is a rank 2 pointed fusion category.