Izumi's group-abelianity conjecture for $3^G$ subfactor planar algebras

Let $G$ be a non-trivial finite group, and let a $3^G$ subfactor planar algebra be a subfactor planar algebra whose principal graph is a $3^{|G|}$ spoke graph. Let $\theta$ be the automorphism of $G$ determined by the fusion rule $\rho\otimes g=\theta(g)\rho$. Izumi's conjecture. If a $3^G$ subfactor planar algebra exists, then $G$ is abelian, and $\theta(g)=g^{-1}$ for all $g\in G$.

This conjecture constrains the possible group and fusion rules underlying these subfactor planar algebras. The source gives no resolution of this statement.