Special Equivariant Plane Conjecture

Let $G$ be a finitely generated group of covering translations acting on a Whitehead manifold $W$, with quotient a 3-manifold $M$. A proper plane $\Pi$ in $W$ is equivariant if, for every $g\in G$, either $g(\Pi)=\Pi$ or $\Pi\cap g(\Pi)=\varnothing$. A plane is non-trivial if no component of its complement has closure homeomorphic to $\mathbf{R}^2\times[0,\infty)$ with the plane corresponding to $\mathbf{R}^2\times\{0\}$. Special Equivariant Plane Conjecture. If $G$ is not a free product of infinite cyclic groups and infinite closed surface groups, then $W$ contains a non-trivial equivariant plane. The source introduces this as a conjecture related to the Universal Covering Conjecture and gives no resolution.