  Astrophysical tests of the stability of Nature's fundamental couplings are a
key probe of the standard paradigms in fundamental physics and cosmology. In
this report we discuss updated constraints on the stability of the
fine-structure constant $\alpha$ and the proton-to-electron mass ratio
$\mu=m_p/m_e$ within the Galaxy. We revisit and improve upon the analysis by
Truppe {\it et al.} by allowing for the possibility of simultaneous variations
of both couplings and also by combining them with the recent measurements by
Levshakov {\it et al.} By considering representative unification scenarios we
find no evidence for variations of $\alpha$ at the 0.4 ppm level, and of $\mu$
at the 0.6 ppm level; if one uses the Levshakov bound on $\mu$ as a prior,
the$\alpha$ bound is improved to 0.1 ppm. We also highlight how these
measurements can constrain (and discriminate among) several fundamental physics
paradigms.
