Let $\ \mathscr U:=(U,\delta)\ $ be a separable metric space which is universal for all finite metric spaces, i.e. for every finite metric space $ \mathscr X:=(X,d)\ $ there
exists an isometric embedding of $\ \mathscr X\ $ into
$\ \mathscr U.$

**Q:**   Does there exist a 0-dimensional subset
$\ C\subset U\ $ in $\ \mathscr U\ $ such that space
$\ (C,\delta|C\!\times\!C)\ $ is universal for all finite metric spaces?


Similar question hold for

 * the subcategory of the above metric spaces of diameter
      $\le 1;$

 * the subcategory of the above metric spaces which are complete.

>*As long as I know, these questions are **open**.*