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A superconducting splittable quadrupole magnet was designed at Fermilab for use in ILC-style cryomodules, in which the magnet is to be assembled around the beam tube to avoid contaminating the ultraclean superconducting RF beam volume. This quadrupole was built and first tested in a liquid helium bath environment at Fermilab, where its quench and magnetic performance were characterized. The device is intended to be cooled by conduction when installed in cryomodules, so a separate test was made at KEK where an appropriate conduction cooling test facility exists. We present results of the thermal performance of the magnet in the conduction cooling mode, and discuss its excitation characteristics in this operating mode.