  We consider DHOST Ia theory interacting gravitationally with an additional
conventional scalar field minimally coupled to gravity. At the linearized level
of perturbations about cosmological background, we find that in the presence of
a slowly rolling extra scalar field, one of the modes generically propagates at
superluminal speed. This result is valid for any stable cosmological
background. We identify a subclass of DHOST Ia theories in which this
superluminality property is absent, and all modes may propagate (sub)luminally.
We discuss possible implications for the interacting DHOST Ia theories.
