  A strong bound on the properties of axion-like particles (ALPs) has been set
by assuming that ALPs are emitted by the protoneutron star just before the
core-bounce in Galactic core-collapse supernovae, and that these ALPs
subsequently convert to $\gamma$-ray photons which ought to be detected by a
$\gamma$-ray mission. This argument has been applied to supernova 1987A to
derive the bound on the ALP-photon coupling $g_{a \gamma \gamma} \lesssim 5.3
\cdot 10^{- 12} \, {\rm GeV}^{- 1}$ for an ALP mass $m_a \lesssim 4.4 \cdot
10^{- 10} \, {\rm eV}$, and can be applied to the next Galactic supernova to
derive the even stronger bound $g_{a \gamma \gamma} \lesssim 2 \cdot 10^{- 13}
\, {\rm GeV}^{- 1}$ for an ALP mass $m_a \lesssim 10^{- 9} \, {\rm eV}$. We
carefully analyze the considered ALP production mechanism and find that it is
oversimplified to an unacceptable extent. By taking into account the minimal
ingredients required by a realistic analysis, we conclude that the previous
results are doomed to failure. As a consequence, all papers quoting the above
bound should be properly revised. Yet, since we are unable to rule out the
possibility that protoneutron stars emit ALPs, in case a core-collapse
supernova explodes in the Galaxy the $\gamma$-ray satellite missions active at
that time should look for photons possibly coming from the supernova.
