  We revisit the fitting of the Fang-Bernevig-Hu model [Europhys. Lett. 86,
67005 (2009)] to inelastic neutron scattering data carried out by Lipscombe et
al. [Phys. Rev. Lett. 106, 057004 (2011)]. We demonstrate that there are many
quite different parameter choices within the experimentally observed phase
(AFM3) that provide approximately equally good fits to the neutron data. We
note that (a) all of these parameter sets lie very close to a point of
transition between the AFM3 phase and one of its neighbors in the classical
phase diagram, and (b) many of them involve rather large values of the
third-neighbor coupling $J_3$. In light of these observations, we discuss
whether the modeling of FeTe may need revision to allow for a slightly non-AFM3
ground state, orbital as well as spin physics on the Fe sites, the effects of
electron itinerancy, or a combination of these.
