  We consider a conjecture made by Ge, Wang and Wu regarding weighted
Alexandrov-Fenchel inequalities for horospherically convex hypersurfaces in
hyperbolic space (a bound, for some physically motivated weight function, of
the weighted integral of the $k^{\mathrm{th}}$ mean curvature in terms of the
area of the hypersurface). We prove an inequality very similar to the
conjectured one. Moreover, when $k$ is zero and the ambient space has dimension
three, we give a counterexample to the conjectured inequality.
