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A spectroscopic calibration sample that is as representative as possible of the galaxies in the<jats:italic>Euclid<\/jats:italic>weak lensing sample is being collected, selecting galaxies from a self-organizing map (SOM) representation of the galaxy color space. Here, we present the results of a near-infrared<jats:italic>H<\/jats:italic>- and<jats:italic>K<\/jats:italic>-band spectroscopic campaign carried out using the LUCI instruments at the LBT. For a total of 251 galaxies, we present new highly reliable redshifts in the 1.3\u2004\u2264\u2004<jats:italic>z<\/jats:italic>\u2004\u2264\u20041.7 and 2\u2004\u2264\u2004<jats:italic>z<\/jats:italic>\u2004\u2264\u20042.7 ranges. The newly-determined redshifts populate 49 SOM cells that previously contained no spectroscopic measurements and almost twice the occupation numbers of an additional 153 SOM cells. A final optical ground-based observational effort is needed to calibrate the missing cells, in particular in the redshift range 1.7\u2004\u2264\u2004<jats:italic>z<\/jats:italic>\u2004\u2264\u20042.7, which lack spectroscopic calibration. 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