  The muon $g-2$ is a compelling quantity due to the current standing tensions
among the experimental average, data-driven theoretical results, and lattice
results. Matching the final target accuracy of the experiments at Fermilab and
J-PARC will constitute a major challenge for the lattice community in the
coming years. For this reason, it is worthwhile to consider different options
to keep the systematic errors under control. In this proceedings, we discuss
finite-volume effects of the leading Hadron Vacuum Polarization (HVP)
contribution to the muon $g-2$ in the presence of C$^{\star}$ boundary
conditions. When considering isospin-breaking corrections to the HVP,
C$^{\star}$ boundary conditions provide a possible consistent formulation of
$\mathrm{QCD+QED}$ in finite volume. Even though these boundary conditions can
be avoided in the calculation of the leading HVP contribution, we find the
interesting result that they remove the leading exponential finite-volume
correction. In practice, compared to the periodic case, C$^{\star}$ boundary
conditions cut the finite-size effects in half on a lattice of physical size
$M_{\pi}L=4$ and by a factor of almost ten for $M_{\pi}L=8$. We discuss the
origin of this reduction and implications for computational efficiency.
