Classification conjecture for positively curved Alexandrov 4-spaces with circle symmetry

Let $T^1$ act isometrically and effectively on $X^4$, where $X^4$ is a $4$-dimensional, closed, positively curved, orientable Alexandrov space. Classification conjecture. Up to equivariant homeomorphism, $X$ is one of the following spaces:

1. the suspension of a spherical $3$-manifold, with a linear action; or
2. a finite quotient of a weighted complex projective space with a linear action.

The conjecture is motivated by the classification of positively curved Alexandrov spaces with circle symmetry and is stated without the additional local Condition Q hypothesis. The paper identifies Condition Q as the remaining obstacle to proving it.