Unknot linearization conjecture for cyclic actions on the four-sphere

Let $\mathbb Z_n$ act smoothly on $S^4$, and suppose its fixed-point set is an unknotted two-sphere. Unknot linearization conjecture. The action is smoothly equivalent to a linear action. In dimension three, every cyclic action on $S^3$ is smoothly equivalent to a linear action, whereas in dimension four there are nonlinear actions whose fixed-point set is a nontrivial $2$-knot. The knot type of the fixed-point set is the only known invariant of two-sphere-fixing cyclic actions on $S^4$, and the conjecture asserts that the unknotted case is linear.