Countable dense homogeneous filters and the Menger covering property

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.