  Axion-like particles (ALPs) coupled to nucleons might be copiously emitted
from a supernova (SN) core. We extend existing bounds on free-streaming ALPs to
the case in which these are so strongly-interacting with the nuclear matter to
be trapped in the SN core. For strongly-interacting ALPs, we also extend the
bound from the absence of an ALP-induced signal in Kamiokande-II neutrino
detector at the time of SN 1987A. We find that combining the different
arguments, SNe exclude values of ALP-nucleon coupling $g_{aN}\gtrsim10^{-9}$
for ALP masses $m_a\lesssim 1\,\mathrm{MeV}$. Remarkably, in the case of
canonical QCD axion models, the SN bounds exclude all values of $m_a \gtrsim
10^{-2}\,\mathrm{eV}$. This result prevents the possibility for current and
future cosmological surveys to detect any signatures due to hot dark matter QCD
axion mass.
