  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.
