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We study the P → γ γ * (P = π0, η, η') transition form factors by means of the local-duality (LD) version of QCD sum rules. For the case of η and η' , the conventional LD model provides a good description of the existing data. However, for the pion form factor we find a disagreement with recent BABAR results for high Q2, even though the accuracy of the LD approximation is expected to increase with Q2. It remains mysterious why both η and η' form factors to virtual photons, on the one hand, and the π0 form factor, on the other hand, show a qualitatively different behaviour, corresponding to a violation of LD that rises with Q2 in the pion case. In a quantum-mechanical example, we show that, for a bound-state size of about 1 fm, the LD sum rule provides an accurate prediction for the form factor for Q2 larger than a few GeV2. [ABSTRACT FROM AUTHOR]