  Several recent spectroscopic investigations have presented conflicting
results on the existence of Na-rich asymptotic giant branch (AGB) stars in the
Galactic globular cluster M4 (NGC6121). The studies disagree on whether or not
Na-rich red giant branch (RGB) stars evolve to the AGB. For a sample of
previously published HER- MES/AAT AGB and RGB stellar spectra we present a
re-analysis of O, Na, and Fe abundances, and a new analysis of Mg and Al
abundances; we also present CN band strengths for this sample, derived from
low-resolution AAOmega spectra. Following a detailed literature comparison, we
find that the AGB samples of all studies consistently show lower abundances of
Na and Al, and are weaker in CN, than RGB stars in the cluster. This is similar
to recent observations of AGB stars in NGC 6752 and M 62. In an attempt to
explain this result, we present new theoretical stellar evolutionary models for
M 4; however, these predict that all stars, including Na-rich RGB stars, evolve
onto the AGB. We test the robustness of our abundance results using a variety
of atmospheric models and spectroscopic methods; however, we do not find
evidence that systematic modelling uncertainties can explain the apparent lack
of Na- rich AGB stars in M4. We conclude that an unexplained, but robust,
discordance between observations and theory remains for the AGB stars in M 4.
