Point-fixing linearization conjecture for cyclic actions on the four-sphere

Let $\mathbb Z_n$ act smoothly on $S^4$, and suppose the action has exactly two fixed points. Point-fixing linearization conjecture. The action is smoothly equivalent to a linear action. This is the analogue for cyclic actions fixing two points of the proposed linearization statement for actions fixing an unknotted two-sphere. The source presents it among open questions, and no resolution is given.