Absorption of bound supernova axion-like particles by the neutron star weakens the heating limits on their photon coupling

Axion-like particles (ALPs) that couple to protons are produced in a core-collapse supernova, and the slow ones remain bound to the new neutron star. Bose, Ganguly, Kumar, and Raj [arXiv:2610.30255] proposed that the photons from the decay of this bound cloud reheat old neutron stars, and they derived strong limits on the ALP-photon coupling. They neglected the absorption of the ALPs by the star, but a bound ALP on an orbit that enters the star and reaches 50 km crosses it about $6\times 10^{18}$ times over the 333 Myr age of PSR J2144$-$3933. We show that inverse nucleon bremsstrahlung in the cold star absorbs every ALP whose orbit enters it within minutes, long before it decays, so only ALPs born outside the final star on nearly circular orbits survive; they carry $9\times 10^{-9} - 6\times 10^{-5}$ of the heating. We compute the protoneutron-star evolution, the production, the general-relativistic orbits and the absorption ourselves, for twelve protoneutron-star evolutions, three nucleon-bremsstrahlung rates normalized to the same supernova energy loss of light ALPs, and equations of state with and without $Δ(1232)$ isobars. PSR J2144$-$3933 still excludes a window at $m_a\simeq 7.7-196$\ MeV, with a lower edge in the photon coupling $280-700$ times higher than without absorption. Pion conversion and the $Δ(1232)$ resonance, which raise the production of surviving ALPs more than their absorption, widen this window to $m_a\simeq$ $7.7-260$\ MeV. The window depends at similar levels on the bremsstrahlung rate and on the radius of the final star. It requires a direct photon coupling that cancels the one induced by ALP-pion mixing, and existing supernova gamma-ray limits may already cover it.