In a recent paper by Xiu-Li Gao, Zhi-Hui Guo, Zhiguang Xiao, and Zhi-Yong Zhou, Phys. Rev. D 105, 094002 (2022), here referred to as I, $S$-wave $ππ$ scattering phase shifts obtained in a lattice-QCD calculation are analyzed using dispersive $S$-matrix methods. We question the reliability of the conclusion from this analysis that, for a pion mass of 391 MeV, the lattice phases favor the presence of both a $σ$-meson bound state and a nearby virtual state. Our main criticism concerns the neglect of the $S$-wave $K\bar{K}$ channel, which was considered alongside additional $s\bar{s}$ interpolating fields in the lattice computation used by the authors of I and also in typical coupled-channel models. As an illustration, some results from such a recent model are presented as well. Concluding remarks concern possible improvements of the analysis in I as well as further model tests.

1 page, 1 figure. v3: Considerably expanded, with an additional figure and several extra references; version accepted for publication in Phys. Rev. D as a Comment. v4: confusing typo corrected in final paragraph, viz. "chiral-symmetry constraints" should read "crossing-symmetry constraints"