How Viable Is a QCD Axion near 10 MeV?

There has been an attempt to revive the visible QCD axion at the 10 MeV scale assuming that it exclusively couples to the first-generation quarks and the electron. This variant of the QCD axion is claimed to remain phenomenologically viable, partly due to a clever model construction that induces tree-level pion-phobia and exploits uncertainties inherent in the chiral perturbation theory. We confront this model with the cosmological domain wall problem, the quality issue and constraints arising from the electron electric dipole moment. It is also pointed out that the gluon loop-generated axion-top coupling can provide a very large contribution to rare $B$-meson decays, such that the present LHCb data for $B^0 \to K^{*0} e^+ e^-$ rule out the model for the axion mass larger than 30 MeV. There is a strong motivation for pushing the experimental analysis of $B \to K^{(*)} e^+ e^-$ to a lower $e^+ e^-$ invariant mass window, which will conclusively determine the fate of the model, as its contribution to this branching ratio significantly exceeds the Standard Model prediction.