Charge correlations suppress unconventional pairing in the Holstein model

In a recent work by Schrodi $\textit{et al}$. [Phys. Rev. B. $\textbf{104}$, L140506 (2021)], the authors find an unconventional superconducting state with a sign-changing order parameter using the Migdal-Eliashberg theory, including the first vertex correction. This unconventional solution arises despite using an isotropic bare electron-phonon coupling in the Hamiltonian. We examine this claim using hybrid quantum Monte Carlo for a single-band Holstein model with a cuprate-like noninteracting band structure and identical parameters to Schrodi $\textit{et al}$.. Our Monte Carlo results for these parameters suggest that unconventional pairing correlations do not exceed their noninteracting values at any carrier concentration we have checked. Instead, strong charge-density-wave correlations persist at the lowest accessible temperatures for dilute and nearly half-filled bands. Lastly, we present arguments for how vertex-corrected Migdal-Eliashberg calculation schemes can lead to uncontrolled results in the presence of Fermi surface nesting.