Unitarizability of pseudo-unitary fusion categories

Determine whether every pseudo-unitary fusion category can be endowed with a unitary structure, i.e., whether every pseudo-unitary fusion category admits a compatible *-structure making it a unitary fusion category.

Background

The authors show that positive finite tensor categories coincide with pseudo-unitary fusion categories. While unitary fusion categories are known to be pseudo-unitary, the converse—whether pseudo-unitary implies unitary—remains unsettled.

Resolving this question would clarify the relationship between algebraic positivity (via Frobenius–Perron dimensions and pivotal traces) and analytic unitarity in fusion categories, with implications for categorical structures used in quantum algebra and topological phases.