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INFN is developing the prototypes of five corrector magnets, from skew quadrupole to dodecapole, which will equip the high-luminosity interaction regions of the high luminosity-LHC (HL-LHC) at LASA lab (Milano, Italy). These magnets are based on a superferric design, which allows a relatively simple, modular, and easy to construct magnet. This activity takes place within the framework of a collaboration agreement between CERN and INFN. The first prototype, a sextupole, has been tested in 2016. We present here the results of the cold tests of the octupole, including the training performed at 4.2 K and the qualification at 2.17 K, that has been assembled and tested in 2017. We report also on the overall experience gained during constructions and tests and the improvements toward the series production. Other important results concern inductance measurements and quench studies.