The free easy quantum group factor conjecture

Let $G\subset U_N^+$ be an easy quantum group, and let $D$ be the category of partitions from which it comes. Assume that $G$ is free, meaning that

$$D\subset NC.$$

Free easy quantum group factor conjecture. The associated von Neumann algebra $L^\infty(G)$ is a factor.

This conjecture is motivated by the correspondence between liberation of easy quantum groups and intersection of their partition categories with $NC$. It has been verified in a number of key cases, notably for the main examples of easy quantum groups, but no general method is known.