Veech's conjecture on non-negative r-median functions

Let $D$ be a bounded domain in $\mathbb{R}^m$ and let $r$ be an admissible function on $D$, meaning that $r$ is positive and $B_{r(x)}(x)\subset D$ for every $x\in D$. Assume that $r$ is locally bounded away from zero:

$$\inf_{x\in K}r(x)>0$$

for each compact set $K\subset D$. An $r$-median function is a function satisfying the corresponding restricted median property.

Veech's conjecture. Every non-negative, $r$-median function on $D$ is harmonic.

The conjecture was formulated by Veech in 1975 and concerns the extension of restricted mean value characterizations of harmonicity to median functions. The supplied text does not state whether this conjecture has been resolved.