The Katětov construction modified for a T0-quasi-metric space

We discuss the existence and uniqueness of a T0-quasi-metric space qU defined by the following three conditions: (i) qU is bicomplete and supseparable, (ii) every isometry between two finite subspaces of qU extends to an isometry of qU onto itself, and (iii) qU contains an isometric copy of every supseparable T0-quasi-metric space.