$\DeclareMathOperator{\Fix}{\operatorname{Fix}}$I am looking for any reference regarding the following problem:

Problem: Consider a smooth almost-free action of $S^1$ on a smooth sphere $S^n$. Then for all finite subgroups $\mathbb{Z}_m\subseteq S^1$, the fixed point set $\Fix(\mathbb{Z}_m)\subseteq S^n$ is either empty or a rational homology sphere.

This is well-known to be true when the action is linear, or by Smith theory when $m=p^k$ is the power of a prime $p$.

A stronger statement would be that for every action of $\mathbb{Z}_m$ on a sphere $S^n$, the fixed point set $\Fix(\mathbb{Z}_m)$ is a rational homology sphere. What I know is that when $m$ is not the power of a prime, then $\Fix(\mathbb{Z}_m)$ is not necessarily a smooth sphere, but I have not found any indication about its possible (rational) homotopy types.

What boggles me the most is that, I had imagined this problem to either well known to be true, or well known to be false, or extremely interesting – however, I have not found any evidence for any of these.