The inverse mean value property for panharmonic functions

Let $D \subset \mathbb{R}^m$, with $m \geqslant 2$, be a bounded domain with sufficiently smooth boundary, and let $r>0$ satisfy $|B_r|=|D|$. A function $u$ is panharmonic if it belongs to $C^1(\overline D)$ and satisfies the relevant panharmonic equation for some parameter $\mu>0$. Let $n_y$ denote the exterior unit normal to $\partial D$, and let $a^\bullet$ be the spherical-mean coefficient used for panharmonic functions. The inverse mean value conjecture. If there exists $x_0\in D$ and some $\mu>0$ such that

$$\frac{2\mu}{m}a^\bullet(\mu r)u(x_0)=\frac{1}{|\partial D|}\int_{\partial D}\frac{\partial u}{\partial n_y}\,\mathrm{d}S_y$$

for every panharmonic $u\in C^1(\overline D)$, then $D=B_r(x_0)$. This is a flux formulation of an inverse mean value property: the theorem cited in the surrounding discussion is known for the corresponding volume integral, while the boundary-flux version is proposed as a suggested characterization under sufficient boundary regularity; its resolution is not established in the supplied text.