  Astrophysical tests of the stability of dimensionless fundamental couplings,
such as the fine-structure constant $\alpha$, are an area of much increased
recent activity, following some indications of possible spacetime variations at
the few parts per million level. Here we obtain updated constraints on the
Bekenstein-Sandvik-Barrow-Magueijo model, which is arguably the simplest model
allowing for $\alpha$ variations. Recent accurate spectroscopic measurements
allow us to improve previous constraints by about an order of magnitude. We
briefly comment on the dependence of the results on the data sample, as well as
on the improvements expected from future facilities.
