  In this note we present a ZFC construction of a non-meager filter which fails
to be countable dense homogeneous. This answers a question of
Hern\'andez-Guti\'errez and Hru\v{s}\'ak. The method of the proof also allows
us to obtain a metrizable Baire topological group which is strongly locally
homogeneous but not countable dense homogeneous.
