No-neck-pinching conjecture for kissing CMC spheres

Let $M$ be a compact three-dimensional manifold with positive Ricci curvature. Suppose that $S_1$ and $S_2$ are embedded constant mean curvature (CMC) spheres in $M$ that kiss at a point $p$. No-neck-pinching conjecture. The point $p$ cannot be a neck-pinching point. The text notes that this remains unknown in general three-manifolds, while the Euclidean case follows from Alexandrov's uniqueness theorem for embedded CMC surfaces.