Low-weight rotational eigenvector conjecture for $3^G$ subfactor planar algebras

Let $G$ be a finite group, let $\cP_\bullet$ be a $3^G$ subfactor planar algebra, and for each $g\in G\setminus\{1\}$ let $p_g$ be the projection in $\cP_{4,+}$ corresponding to $g\rho$. Suppose $g,h,k,\ell\in G\setminus\{1\}$ are distinct. Low-weight rotational eigenvector conjecture.

1. If $g=g^{-1}$ and $h=h^{-1}$, then $p_g-p_h$ is a low-weight rotational eigenvector with eigenvalue $1$.
2. If $g=h^{-1}$, then $p_g-p_h$ is a low-weight rotational eigenvector with eigenvalue $-1$.
3. If $g=g^{-1}$ and $h=k^{-1}$, then $2p_g-(p_h+p_k)$ is a low-weight rotational eigenvector with eigenvalue $1$.
4. If $g=h^{-1}$ and $k=\ell^{-1}$, then $(p_g+p_h)-(p_k+p_\ell)$ is a low-weight rotational eigenvector with eigenvalue $1$.

These formulas agree with the known Haagerup and Izumi examples and are intended as a first step toward a uniform planar-algebraic approach to $3^G$ subfactors. The source gives no resolution of the conjecture.