Finiteness of Frobenius subalgebra lattices in fusion categories

Prove that for every connected Frobenius algebra object X in any fusion category, the lattice of Frobenius subalgebras of X is finite.

Background

The paper establishes finiteness of the Frobenius subalgebra lattice under positivity assumptions, notably for pseudo-unitary fusion categories, and via semisimplification for spherical tensor categories. These results generalize Watatani’s theorem from subfactor theory to the tensor-categorical setting.

Conjecture 8.10 asks whether this finiteness persists in full generality for fusion categories, beyond the pseudo-unitary or spherical assumptions. A counterexample would require a fusion category not Grothendieck equivalent to a pseudo-unitary one, which the authors note are rare in the literature.