  In this note we show that given two complete geodesic Gromov hyperbolic
spaces that are roughly isometric and $\varepsilon>0$, either the
uniformization of both spaces with parameter $\varepsilon$ results in uniform
domains, or else neither uniformized space is a uniform domain. The terminology
of "uniformization" is from the work of Bonk, Heinonen and Koskela, where it is
shown that the uniformization, with parameter $\varepsilon>0$, of a complete
geodesic Gromov hyperbolic space results in a uniform domain provided
$\varepsilon$ is small enough.
