The simply connected smooth-locus conjecture for rational homology projective planes

Let $S$ be a rational homology $\mathbb C\mathbb P^2$ with quotient singularities, and write

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

Assume that $S^0$ is simply connected. Simply connected smooth-locus conjecture. Then $S$ has at most three singular points. The source also states the stronger related conjecture that $S$ is rational, and notes that this stronger statement was verified by Keum when the singularities are not very complicated; the supplied source does not resolve the three-singular-point bound.