The algebraic Montgomery–Yang problem for rational homology projective planes

Let $S$ be a rational homology projective plane, meaning a normal projective complex surface whose Betti numbers are the same as those of $\mathbb{CP}^2$. Assume that $S$ has quotient singularities and write

$$S^0:=S\setminus\operatorname{Sing}(S).$$

If $S^0$ is simply connected, then the algebraic Montgomery–Yang problem. $S$ has at most $3$ singularities.

This is a special case of Kollár's classification problem for rational homology projective planes with quotient singularities. The conjecture has been confirmed by Hwang and Keum, except when $S$ is a rational surface with an ample canonical divisor and only cyclic singularities.