The Kac type characterization conjecture for second countable compact quantum groups

Let $\mathbb{G}$ be a second countable compact quantum group. The invariant $\mathnormal{T}^{\tau}_{\operatorname{Inn}\!}(\mathbb{G})$ is defined in the paper's preceding discussion.

Kac type characterization conjecture. If

$$\mathnormal{T}^{\tau}_{\operatorname{Inn}\!}(\mathbb{G})=\mathbb{R},$$

then $\mathbb{G}$ is of Kac type.

This conjecture asks whether the condition that the indicated invariant equals $\mathbb{R}$ characterizes Kac type among second countable compact quantum groups. The paper presents a positive answer for a broad class of compact quantum groups, while the general case remains open.