Two-box generation conjecture for reduced $3^G$ subfactor planar algebras

Let $G$ be a non-trivial finite group, let $\cR_\bullet$ be the reduced subfactor planar algebra of a $3^G$ subfactor planar algebra, and let $\cQ_\bullet$ be the planar subalgebra of $\cR_\bullet$ generated by the projections $p_g$ for $g\ne 1$ together with the specified two-box element, under the $\rho$-strand planar operad whose tangles contain no trivalent vertices. Two-box generation conjecture.

$$\cQ_\bullet=\cR_\bullet.$$

The authors describe this as highly non-trivial and prove it when $|G|$ is odd. The assertion remains open for general $|G|$ in the source.