$\require{AMScd}$

In the paper [1], it is shown that there exist finite dimensional, semisimple Hopf algebras $H$ where the antipode $S:H \to H$ is traceless.

Unfortunately, the method of proof in [1] seems rather inexplicit to me. My understanding is that the authors use the fact that every fusion category $\mathcal{C}$ satisfying some conditions is the representation category of some Hopf algebra. Then they convert the trace condition $\text{Tr}(S) = 0$ into a condition on the fusion category $\mathcal{C}$ and show that they can easily construct categories satisfying this new condition, in addition to the conditions necessary for $\mathcal{C}$ to be a representation category of some $H$.

The paper [1] is eight years old now, so I'm hoping that some more explicit examples might be in the literature at this point. So my question is:

Question:What are some explicit examples of a finite dimensional, involutory Hopf algebra (over a field $k$ of characteristic $0$) where the antipode is traceless? By explicit, I mean given by a generator/relation presentation or as some structure tensors in a specific basis.

Here I weakened the hypothesis to assume that $H$ is just involutory, i.e. that $S^2 = \text{Id}$. A theorem of Larson and Radford [2] says that semisimple Hopf algebras are involutory when $\text{char}(k) = 0$.

Alternatively, I would appreciate some advice on what I need to know in order to extract an explicit example from [1], if that is what's necessary.