Strong Green function rigidity conjecture for embedded hypersurfaces

Let $M^n\subset\mathbb{R}^{n+1}$ be a smooth closed embedded hypersurface. Let $p\in M$ be a point such that the Green function of the corresponding conformally invariant operator has the form described for the round sphere, namely an expression in terms of the Euclidean distance from $p$ to another point of $M$.

Strong Green function rigidity conjecture. If there exists such a point $p\in M$, then $M$ must be a round sphere.

This is a converse to the explicit Green function formulas for conformally invariant operators on round spheres. The stronger condition that the formula hold for every pair of points implies that all points of $M$ are umbilical; the conjecture asks whether the condition at just one point already forces spherical geometry.