For Alexandrov manifold in the title we mean 3-dim Alexandrov apace which is also a topological. manifold. Shioya-Yamaguchi posted a conjecture on their paper "Collapsing 3-manifold with lower sectional curvature bound"
Any three-dimensional compact, simply connected, nonnegatively curved Alexandrov space without boundary which is a topological manifold is homeomorphic to a sphere.
It seems for me that Riemannian geometric tools do not apply here. So any progress in this directoin or it has been proved somewhere?