  Supernova remnant (SNR) G106.3+2.7 was recently found to be one of the few
potential Galactic hadronic PeVatrons. Aiming to test how solid the SNR is
associated with the molecular clouds (MCs) that are thought to be responsible
for hadronic interaction, we performed a new CO observation with the IRAM 30m
telescope toward its "belly" region, which is coincident with the centroid of
the $\gamma$-ray emission. There is a filament structure in the
local-standard-of-rest velocity interval $-8$ to $-5$ km/s that nicely follows
the northern radio boundary of the SNR. We have seen asymmetric broad profiles
of $^{12}$CO lines, with widths of a few km/s along the northern boundary and
in the "belly" region of G106.3+2.7, but similar $^{12}$CO line profiles are
also found outside the SNR boundary. Further, the low $^{12}$CO J=2-1/J=1-0
line ratios suggest the MCs are cool. Therefore, it is still uncertain whether
the MCs are directly disturbed by the SNR shocks, but we do find some clues
that the MCs are nearby and thus can still be illuminated by the escaped
protons from the SNR. Notably, we find an expanding molecular structure with a
velocity of $\sim$3.5 km/s and a velocity gradient of the MCs across the SNR
from $\sim -3$ to $-7$ km/s, which could be explained as the effect of the wind
blown by the SNR's progenitor star.
