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

Galactic planetary nebulae with precise nebular abundances as a tool to understand the evolution of asymptotic giant branch stars