General Topology – Is a Uniform Group Paracompact?

In this work Martin Schottenloher notes that the unified group $ U (H) $ a separable Hilbert room $ H $ is measurable in the strong operator topology, As a consequence (see R. Engelking, 5.1.3) it is paracompact (if $ H $ is separable). I wonder

if $ U (H) $ is paracompact for any (not necessarily separable) Hilbert space $ H $ in that strong operator topology,

P.S. Maybe this is a known fact, excuse my ignorance in this case. I think this will continue the discussions here, here, here and here, but I have to say that I do not even understand why $ U (H) $ is not locally compact in the infinite dimensional case.