The linearisation conjecture for finite group actions on the 4-sphere

Let $G$ be a finite group acting smoothly on $\mathbb S^4$, and suppose the action has an isolated fixed point. The action is linear when it is smoothly conjugate to a linear action on the sphere.

Linearisation conjecture. Every smooth finite group action on $\mathbb S^4$ with an isolated fixed point is smoothly conjugate to a linear action.

The question reduces, after removing an invariant neighborhood of the fixed point, to the corresponding problem for a smooth action on the $4$-ball that is free and linear on the boundary. The preceding discussion relates it to the symplectic $s$-cobordism problem for elliptic $3$-manifolds; no resolution is supplied in the source.