The four-object fusion category conjecture for cases (a)–(c)

Consider the three fusion-category cases (a)–(c) described immediately before this conjecture. The relevant objects are invertible objects and objects of Frobenius–Perron dimension $4+\sqrt{17}$, and the Brauer–Picard group is the group of Morita autoequivalences under consideration.

Fusion category classification conjecture. Cases (a)–(c) are each realized by a unique fusion category, and each such category has the following properties: it has $4$ invertible objects and $4$ objects of dimension $4+\sqrt{17}$, and its Brauer–Picard group is implemented by outer automorphisms.

This is presented as a further conjectural refinement of the preceding classification theorem. The source supplies no resolution of the uniqueness and outer-automorphism assertions.