NICE candidates selection criteria. Left: IRAC color to constrain the redshift to be 2 ≲ z ≲ 4. Data points are UVJ color selected massive SFGs from COSMOS2020 (Weaver et al. 2022). Solid and dashed red curves show the median and the lower limit of 90% of the galaxies. The yellow shade shows the selection criterion. Middle: IRAC2 magnitude to choose galaxies more massive than 10¹⁰ M_⊙. Data points are UVJ color selected SFGs at 2 < z_phot < 4 from COSMOS2020. The red curves and the yellow shade are the same as in the left panel. Right: Herschel/SPIRE colors to select starbursting structures at 2 ≲ z ≲ 4. Grey dots are the Herschel detections in all the NICE survey fields, while the black dots meet the selection criteria in Eq. (2). The blue, red, and magenta curves show the trend of main sequence galaxies, starbursts and SMGs and the circles are marked by the redshifts accordingly. The starbursting proto-cluster at z = 2.51, CL-J1001, is presented as an orange star. See details in Sect. 2.2.2. The inserted histogram displays the distribution of 500 μm fluxes.