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Recently, the Large High Altitude Air Shower Observatory (LHAASO) identified12 $\gamma$-ray sources emitting gamma rays with energies above 100 TeV, makingthem potential PeV cosmic-ray accelerators (PeVatrons). Neutrino observationsare crucial in determining whether the gamma-ray radiation process is ofhadronic or leptonic origin. In this paper, we study three detected sources,LHAASO J1908+0621, LHAASO J2018+3651, and LHAASO J2032+4102, which are also themost promising galactic high-energy neutrino candidate sources with the lowestpre-trial p-value based on the stacking searches testing for excess neutrinoemission by IceCube Neutrino Observatory. We study the lepto-hadronic scenariofor the observed multiband spectra of these LHAASO sources considering thepossible counterpart source of the LHAASO sources. The very-high-energy gammarays are entirely attributed to the hadronic contribution, therefore the mostoptimistic neutrino flux can be derived. Then, we evaluate the statisticalsignificance (p-value) as a function of the observation time of IceCube and thenext-generation IceCube-Gen2 neutrino observatory respectively. Our resultstend to disfavor that all gamma rays above $100\,\rm GeV$ from LHAASOJ1908+0621 are of purely hadronic origin based on current IceCube observations,but the purely hadronic origin of gamma rays above $100\,\rm TeV$ is stillpossible. By IceCube-Gen2, the origin of gamma rays above $100\,\rm TeV$ fromLHAASO J1908+0621 can be further determined at a $5\sigma$ significance levelwithin a running time of $\sim 3$ years. For LHAASO J2018+3651 and LHAASOJ2032+4102, the required running time of IceCube-Gen2 is $\sim 10$ years($3\sigma$) and $\sim 10$ years ($5\sigma$), respectively. Future observationsby the next-generation neutrino telescope will be crucial to understanding theparticle acceleration and radiation processes inside the sources.

Determining the Origin of Very-high-energy Gamma Rays from Galactic Sources by Future Neutrino Observations