Comment on "Chiral symmetry restoration, the eigenvalue density of the Dirac operator, and the axial U(1) anomaly at finite temperature"

Aoki, Fukaya, and Taniguchi claim that both the spectral density of the Dirac operator at the origin and the topological susceptibility must vanish identically for sufficiently small but nonzero mass $m$ in the chirally symmetric phase of QCD with two light quark flavors, under certain technical assumptions on the spectrum and on the dependence of observables on $m$. Independently of these assumptions, I argue that a crucial step of their proof is not justified, and the validity of these conclusions should be reassessed.