A Median graph is graph with the property, that for each three vertices $x,y,z$ there is a unique vertex $m(x,y,z)$ lying on shortest paths from $x$ to $y$, from $y$ to $z$ and from $z$ to $x$. Examples are trees, the Cayley graph of $\mathbb{Z}^n$ (with the standart generating set) and cross products of other median graphs.

Suppose, that $x$ and $x'$ are connected by an edge. Is it true, that $m(x,y,z)$ and $m(x',y,z)$ are also connected by an edge ?