Weighted logarithmic mean characterisation of harmonic functions

Let $D$ be a bounded domain in $\mathbb{R}^m$, with $m\geqslant 2$. Let $A_x(r_1,r_2)$ be an admissible annular domain centred at $x$, with $r_2>r_1>0$, and let $B_{r_2}(x)$ be admissible. Denote by $M^\circ(S_r(x),u)$ the spherical mean of $u$ over $S_r(x)$, and let $r$ either attain each value in $(r_1,r_2)$ or equal $r_i$ for $i=1$ or $2$. Weighted logarithmic mean conjecture. If $u\in C^0(D)$ satisfies, for every $x\in D$ and every such $A_x(r_1,r_2)$,

$$\frac{m}{|A(r_1,r_2)|}\int_{A_x(r_1,r_2)}u(y)\log\frac{r}{|x-y|}\,\mathrm{d}y=M^\circ(S_r(x),u),$$

then $u$ is harmonic in $D$. The identity extends the logarithmic weighted mean property of harmonic functions from balls to annular domains; the stated result is presented after the corresponding harmonic characterisations and its resolution is not established in the supplied text.