Are there examples of finite-dimensional weak Hopf C*-algebras with non-involutive antipode?

According to this paper of Nikshych-Vainerman, a finite-dimensional weak Kac algebra is precisely a finite-dimensional weak Hopf ${\rm C}^{\star}$-algebra with an involutive antipode ($S^2 = id$), and they give (p306) an explicit example of finite-dimensional weak Hopf ${\rm C}^{\star}$-algebra which is not a weak Kac algebra.

More generally, any finite index depth $2$ unital inclusion of type ${\rm II}_1$ factors is characterized by a finite-dimensional weak Hopf ${\rm C}^{\star}$-algebra and in the weak Kac case, the index is an integer. So every non-integer finite index depth $2$ subfactor provides an example of finite-dimensional weak Hopf ${\rm C}^{\star}$-algebra with a non-involutive antipode. For any depth $n$ subfactor $(N \subset M)$ of finite index $|M:N|$, if $$N \subset M \subset M_1 \subset M_2 \subset M_3 \subset \cdots $$ is the Jones tower, then $(N \subset M_{n-2})$ is a depth $2$ subfactor of index $|M:N|^{n-1}$. The depth $n$ Temperley-Lieb-Jones subfactor has index $4cos^2(\frac{\pi}{n+1})$, and $[4cos^2(\frac{\pi}{n+1})]^{n-1}$ is not an integer for $n \neq 2,4$, so the corresponding weak Hopf ${\rm C}^{\star}$-algebra has a non-involutive antipode (it should also be the case for $n=4$, but not for $n=2$). The example of Nikshych-Vainerman above is $n=3$.

---

This paper of Das proves by planar algebra techniques that for any finite index depth $2$ subfactor $(N \subset M)$, the relative commutants $N' \cap M_1$ and $M' \cap M_2$ has a structure of finite-dimensional weak Hopf ${\rm C}^{\star}$-algebra. What does the diagram on p84 say about the antipode $S$?

This is an answer to the question in the OPs answer.

I just took a brief look into

Rehren - Weak C Hopf symmetry* in Quantum Groups Symposium at "Group21", eds. H.-D. Doebner et al., Goslar 1996 Proceedings, Heron Press, Sofia (1997), pp. 62-69. https://arxiv.org/abs/q-alg/9611007

which does the type III case.

The point is that $S^{-1}(q)=S^\ast(q^\ast)$, which is what the diagram on p84 in the mentioned article says.