Cylindrical ball neighborhood conjecture for isolated critical points

Let $f\colon M\to\mathbb{R}$ be a smooth function on a manifold $M$ without boundary, and let $x_0\in M$ be an isolated critical point. A cylindrical ball neighborhood is a cylindrical neighborhood of $x_0$ that is homeomorphic to a ball. Cylindrical ball neighborhood conjecture. Every isolated critical point of any smooth function admits a cylindrical ball neighborhood. This would provide cylindrical ball neighborhoods as the local building blocks needed for Lusternik–Schnirelmann-type inequalities; the source presents the claim as a conjecture and gives evidence for it, but does not state a resolution.