The core-resolution conjecture for positively curved G-manifolds

Let $M$ be a positively curved $G$-manifold with principal isotropy group $H\ne\{1\}$. Write ${}^r\!M$ for the core resolution of $M$. Core-resolution conjecture. Either

$$M=G/H,$$

or

$${}^r\!M\ne M.$$

In particular, $M$ has singular $G$-orbits. The conjecture concerns the rigidity of positively curved $G$-manifolds with nontrivial principal isotropy: apart from the homogeneous case, the core resolution should be nontrivial, forcing the existence of singular orbits.