We refer to [JS97] for the notion of subfactor. Yasuo Watatani proved the following result [W96, Theorem 2.2]:

Theorem: An irreducible finite index subfactor of a type $\mathrm{II}_1$ factor has finitely many intermediate subfactors.

We refer to [EGNO15] for the notion of tensor category. We are interested in extending Watatani's theorem to tensor categories. Dave Penneys reformulated it in this framework as follows (private communication):

Theorem: A connected unitary Frobenius algebra in a unitary tensor category has finitely many unitary Frobenius subalgebras.

Question: How to prove above Theorem directly in the tensor category framework? Is the unitary assumption required?

Remark: It is also proved in [W96] that the set of intermediate subfactors forms a lattice. So we could also check whether the set of Frobenius subalgebras always forms a lattice.

A stronger version of Watatani's theorem was proved in [BDLR19], providing a bound for the cardinal of the lattice. Its proof involves a new notion of angle between two intermediate subfactors and Pimsner-Popa basis. Note that the arXiv version of [BDLR19] also contains a purely planar algebraic (short) proof only involving the notion of angle between two biprojections. Thus, a generalization of the notion of angle to between two Frobenius subalgebras may help answer the above question.

References

[BDLR19] Bakshi, Keshab Chandra; Das, Sayan; Liu, Zhengwei; Ren, Yunxiang. An angle between intermediate subfactors and its rigidity. Trans. Amer. Math. Soc. 371 (2019), no. 8, 5973--5991; arXiv:1710.00285.
[EGNO15] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. Tensor categories. Mathematical Surveys and Monographs, 205. American Mathematical Society, Providence, RI, 2015. xvi+343 pp.
[JS97] Jones, V.; Sunder, V. S. Introduction to subfactors. London Mathematical Society Lecture Note Series, 234. Cambridge University Press, Cambridge, 1997. xii+162 pp.
[W96] Watatani, Yasuo. Lattices of intermediate subfactors. J. Funct. Anal. 140 (1996), no. 2, 312--334.