Radio Proper Motions of the Nearby Ultra-cool dwarf binary VHS 1256$-$1257AB

The proper motions of a source obtained at different epochs or in different spectral regions should in principle be consistent. However, in the case of a binary source or a source with associated ejecta, they could be different depending on the epochs when the observations were made and on what emission is traced in each spectral region. In this paper we determine the radio proper motions of the ultra-cool dwarf binary VHS 1256$-$1257AB from Very Large Array (VLA) observations, that we find are consistent within error ($\simeq 2-3\%$) with those reported by Gaia DR3. The comparison of the proper motions and the analysis of the VLA data imply that, as in the optical, the radio emission is coming in comparable amounts from both components of the unresolved binary.