  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 $\pi\pi$
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 $\sigma$-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.

High Energy Physics - Phenomenology;High Energy Physics - Lattice