  X-ray reflection spectroscopy can be a powerful tool to test the nature of
astrophysical black holes. Extending previous work on Kerr black holes with
scalar hair [1] and on boson stars [2], here we study whether astrophysical
black hole candidates may be horizonless, self-gravitating, vector
Bose-Einstein condensates, known as Proca stars [3]. We find that observations
with current X-ray missions can only provide weak constraints and rule out
solely Proca stars with low compactness. There are two reasons. First, at the
moment we do not know the geometry of the corona, and therefore the uncertainty
in the emissivity profile limits the ability to constrain the background
metric. Second, the photon number count is low even in the case of a bright
black hole binary, and we cannot have a precise measurement of the spectrum.
