  We present nucleosynthesis predictions (HeCNOCl) from asymptotic giant branch
(AGB) models, with diffusive overshooting from all the convective borders, in
the metallicity range Z/4 < Z < 2Zsun. They are compared to recent precise
nebular abundances in a sample of Galactic planetary nebulae (PNe) that is
divided among double-dust chemistry (DC) and oxygen-dust chemistry (OC)
according to the infrared dust features. Unlike the similar subsample of
Galactic carbon-dust chemistry PNe recently analysed by us, here the individual
abundance errors, the higher metallicity spread, and the uncertain dust
types/subtypes in some PNe do not allow a clear determination of the AGB
progenitor masses (and formation epochs) for both PNe samples; the comparison
is thus more focussed on a object-by-object basis. The lowest metallicity OC
PNe evolve from low-mass (~1 Msun) O-rich AGBs, while the higher metallicity
ones (all with uncertain dust classifications) display a chemical pattern
similar to the DC PNe. In agreement with recent literature, the DC PNe mostly
descend from high-mass (M > 3.5 Msun) solar/supersolar metallicity AGBs that
experience hot bottom burning (HBB), but other formation channels in low-mass
AGBs like extra mixing, stellar rotation, binary interaction, or He
pre-enrichment cannot be disregarded until more accurate C/O ratios would be
obtained. Two objects among the DC PNe show the imprint of advanced CNO
processing and deep second dredge-up, suggesting progenitors masses close to
the limit to evolve as core collapse supernovae (above 6 Msun). Their actual
C/O ratio, if confirmed, indicate contamination from the third dredge-up,
rejecting the hypothesis that the chemical composition of such high-metallicity
massive AGBs is modified exclusively by HBB.
