What are the module categories over the modular tensor category Fib of Fibonacci anyons?
By Ostrik's work, we know these module categories correspond to separable algebras in Fib. I do not believe such things have been classified.
Davydov and Booker show there are no nontrivial commutative separable algebras in Fib, but I do not see that they make a clear statement for this more general case, without commutativity.
My guess is that there are indeed no nontrivial module categories for Fib.