We summarize the evidence for DM-like anomalies in neutron multiplicity spectra collected underground with Pb targets by three independent experiments: NEMESIS (at 210 m.w.e.) NMDS (at 583 m.w.e.), and ZEPLIN-II (at 2850 m.w.e.). A new analysis shows small but persistent anomalies at high neutron multiplicities. Adjusted for differences in detection efficiencies, the positions of the anomalies are consistent between the three systems. Also, the intensities match when corrected for the acquisition time and estimated detection efficiency. While the three measurements are inconclusive when analyzed separately, together, they exclude a statistical fluke to better than one in a million. To prove the existence of the anomalies above the 5-sigma discovery threshold, we propose to upgrade the current NEMESIS setup. The upgrade concept and the critical components of the new experiment are described. The upgraded setup would already acquire the needed data sample during the first year of operation. Additional information, vital for the physics interpretation of the analysis, will be obtained with a Cu target.