Green function rigidity conjecture for closed hypersurfaces

Let $M^n\subset\mathbb{R}^{n+1}$ be a closed hypersurface with induced Riemannian metric $g$. For some $k\in\mathbb{Z}_+$, suppose the Green function $G$ of the GJMS operator $P_{2k}^g$ exists. Assume one of the following conditions holds:

1. $2k=n$ and, for some $Q\in M$, $G(\cdot,Q)$ has the form

$$-c_n\log\|\cdot-Q\|+c.$$

2. $2k<n$, or $2k>n$ when $n$ is odd, and, for some $Q\in M$, $G(\cdot,Q)$ has the form

$$c_{n,k}\|\cdot-Q\|^{2k-n}.$$

Green function rigidity conjecture. Then $(M,g)$ is a round sphere.

The preceding rigidity theorem proves this conclusion for ordinary GJMS operators of orders up to $4$ under stronger pointwise assumptions involving every pair of points $P,Q\in M$. The conjecture asks whether the corresponding Green-function forms for a single pole characterize the round sphere for the broader range of orders stated above.