Recent studies of heavy $J/\psi$ and $D$-mesons production by the ALICE and STAR collaborations revealed a pronounced dependence of the cross-section on multiplicity of co-produced charged particles. One of the possible explanations of this phenomenon is the enhanced contribution of multipomeron configurations. We suggest that the study of the multiplicity dependence of $P$-wave quarkonia (e.g. $\chi_{c}$ and $\chi_{b}$ mesons) could provide decisive evidence in favor of this explanation. Due to different quantum numbers, the $P$-wave quarkonia will not get contributions from the three-pomeron mechanism, and for this reason should have significantly milder dependence on multiplicity. We also present detailed production cross-sections in the kinematics of ongoing experiments at LHC and RHIC.