  Weak G-band (wGb) stars are very peculiar red giants almost devoided of
carbon and often mildly enriched in lithium. Despite their very puzzling
abundance patterns, very few detailed spectroscopic studies existed up to a few
years ago, preventing any clear understanding of the wGb phenomenon. We
recently proposed the first consistent analysis of published data for 28 wGb
stars and identified them as descendants of early A-type to late B-type stars,
without being able to conclude on their evolutionary status or the origin of
their peculiar abundance pattern.
  We used newly obtained high-resolution and high SNR spectra for 19 wGb stars
in the southern and northern hemisphere to homogeneously derive their
fundamental parameters, metallicities, as well as the spectroscopic abundances
for Li, C, N, O, Na, Sr, and Ba. We also computed dedicated stellar evolution
models that we used to determine the masses and to investigate the evolutionary
status and chemical history of the stars in our sample. We confirm that the wGb
stars are stars in the mass range 3.2 to 4.2 M$_\odot$. We suggest that a large
fraction could be mildly evolved stars on the SGB currently undergoing the 1st
DUP, while a smaller number of stars are more probably in the core He burning
phase at the clump. After analysing their abundance pattern, we confirm their
strong N enrichment anti-correlated with large C depletion, characteristic of
material fully processed through the CNO cycle to an extent not known in other
evolved intermediate-mass stars. However, we demonstrate here that such a
pattern is very unlikely due to self-enrichment. In the light of the current
observational constraints, no solid self-consistent pollution scenario can be
presented either, leaving the wGb puzzle largely unsolved.
