The cyclic fundamental group conjecture for unknotted surfaces

Let $\mathcal{K}$ be a knotted surface in $S^4$, and let $S^4\setminus\mathcal{K}$ be its exterior. The fundamental group of the exterior is $\pi_1(S^4\setminus\mathcal{K})$. Cyclic fundamental group conjecture. The surface $\mathcal{K}$ is unknotted if and only if $\pi_1(S^4\setminus\mathcal{K})$ is cyclic. This is a central conjecture about recognizing unknotted surfaces from their knot groups; its resolution is not supplied here.