Bose-Einstein condensates (BECs) have been proposed as candidate states of matter for the interior of neutron stars. Specifically, Chavanis and Harko obtained the mass-radius relation for a BEC star and proposed that the recently discovered neutron stars with masses around 2$M_odot$ are BEC stars. They employed a barotropic equation of state (EOS), with one free parameter, that was first found by Colpi, Wasserman, and Shapiro (CSW), to describe them and derive stable equilibrium configurations of spinning BEC stars in General Relativity. In this work we show that while it is true that BECs allow for compact object masses as heavy as the heaviest observed ones, such stars cannot simultaneously have radii that are small enough to be consistent with the latest observations, in spite of the flexibility available in the EOS in the form of the free parameter. In fact, our conclusion applies to any spinning relativistic boson star that obeys the CSW EOS.
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