  Recently, the Large High Altitude Air Shower Observatory (LHAASO) identified
12 $\gamma$-ray sources emitting gamma rays with energies above 100 TeV, making
them potential PeV cosmic-ray accelerators (PeVatrons). Neutrino observations
are crucial in determining whether the gamma-ray radiation process is of
hadronic or leptonic origin. In this paper, we study three detected sources,
LHAASO J1908+0621, LHAASO J2018+3651, and LHAASO J2032+4102, which are also the
most promising galactic high-energy neutrino candidate sources with the lowest
pre-trial p-value based on the stacking searches testing for excess neutrino
emission by IceCube Neutrino Observatory. We study the lepto-hadronic scenario
for the observed multiband spectra of these LHAASO sources considering the
possible counterpart source of the LHAASO sources. The very-high-energy gamma
rays are entirely attributed to the hadronic contribution, therefore the most
optimistic neutrino flux can be derived. Then, we evaluate the statistical
significance (p-value) as a function of the observation time of IceCube and the
next-generation IceCube-Gen2 neutrino observatory respectively. Our results
tend to disfavor that all gamma rays above $100\,\rm GeV$ from LHAASO
J1908+0621 are of purely hadronic origin based on current IceCube observations,
but the purely hadronic origin of gamma rays above $100\,\rm TeV$ is still
possible. By IceCube-Gen2, the origin of gamma rays above $100\,\rm TeV$ from
LHAASO J1908+0621 can be further determined at a $5\sigma$ significance level
within a running time of $\sim 3$ years. For LHAASO J2018+3651 and LHAASO
J2032+4102, the required running time of IceCube-Gen2 is $\sim 10$ years
($3\sigma$) and $\sim 10$ years ($5\sigma$), respectively. Future observations
by the next-generation neutrino telescope will be crucial to understanding the
particle acceleration and radiation processes inside the sources.
