Let $\mathbb{G}$ be a discrete quantum group, and consider the non-commutative Furstenberg boundary $\partial_F\mathbb{G}$ with function algebra $C(\partial_F \mathbb{G}) = I_{\mathbb{G}}(\mathbb{C})$. Here, $I_\mathbb{G}(\mathbb{C})$ is injective so has a $C^*$-algebra structure coming from the Choi-Effros product.

Is something known about nuclearity of this $C^*$-algebra? Of course, if $\mathbb{G}$ is a classical discrete group, then $C(\partial_F\mathbb{G})$ is commutative, so nuclear. Are there non-trivial examples known of genuine discrete quantum groups where this $C^*$-algebra is nuclear or not?