  We discuss the future prospects of a minimally five-dimensional version of
the well-motivated scenario for addressing the discrepancy in the muon
anomalous magnetic moment, the $U(1)_{L_\mu - L_\tau}$ extension of the
standard model (SM) gauge symmetry. Here, multiple associated massive gauge
bosons appear thanks to the five-dimensional $U(1)_{L_\mu - L_\tau}$ gauge
symmetry, and they contribute to the muon $(g-2)$ and also other processes. We
focus on the powerful probe of elastic neutrino-electron scatterings since the
upcoming DUNE experiment will explore MeV-scale uncharted regions by previous
experiments (e.g., CHARM-II and Borexino) in the near future. We found that
even with small kinetic mixing parameters, much of the parameter space,
including those satisfying muon $(g-2)$, can be probed using several years of
data from the DUNE experiment, focusing on the near detector. In our scenario,
interference effects between intermediate-state gauge bosons play an important
role. Our results include comparisons between flat and warped extra dimensions.
