The search for the existence of a nonzero neutron electric dipole moment (nEDM) has the potential to reveal new sources of T and CP violation beyond the Standard Model and may have a significant impact on our understanding of the universe. A new experiment aiming at two orders of magnitude improvement ($\sim10^{-28}$ e$\cdot$cm) over the current experimental upper limit has been proposed in the United States. In the experiment, the measurement cell will be made of dTPB-dPS coated acrylic and filled with superfluid $^4$He at $\sim$300-500 mK. The measurement of the neutron precession frequency will rely on the spin-dependence of the cross section of the nuclear reaction between polarized neutrons and $^3$He atoms: $\vec{n}+\vec{^{3}He}\rightarrow p+t+764$ keV. Polarized $^3$He will also be used as a comagnetometer based on the nuclear magnetic resonance technique. The $^3$He polarization needs to have sufficiently long relaxation time so that little polarization is lost during the measurement period in order to achieve the proposed sensitivity. Understanding the relaxation mechanism of $^3$He polarization in the measurement cell under the nEDM experimental conditions and maintaining $^3$He polarization is crucial for the experiment.