  The substellar triple system VHS J125601.92$-$125723.9 is composed by an
equal-mass M7.5 brown dwarf binary and a L7 low-mass substellar object. In this
work, we aim to identify the origin of the radio emission occurring in the
central binary of VHS 1256$-$1257 while discussing the expected mechanisms
involved in the radio emission of ultracool dwarfs (UCDs). We observed this
system with the Karl G. Jansky Very Large Array, the European
very-long-baseline interferometry (VLBI) Network, the enhanced Multi Element
Remotely Linked Interferometer Network, the NOrthern Extended Millimeter Array,
and the Atacama Large Millimetre Array at frequencies ranging from 5 GHz up to
345 GHz in several epochs during 2017, 2018, and 2019. We have found radio
emission at 6 GHz and 33 GHz coincident with the expected position of the
central binary of VHS~1256$-$1257. The Stokes I density flux detected were 73
$\pm$ 4 $\mu$Jy and 83 $\pm$ 13 $\mu$Jy, respectively, with no detectable
circular polarisation or pulses. No emission is detected at higher frequencies
(230 GHz and 345 GHz) nor at 5 GHz with VLBI arrays. The emission appears to be
stable over almost 3 years at 6 GHz. To explain the constraints obtained both
from the detections and non-detections we considered multiple scenarios
including thermal and non-thermal emission, and different contributions from
each component of the binary. Our results can be well explained by non-thermal
gyrosynchrotron emission originating at radiation belts with a low plasma
density (n$_e$ = 300$-$700 cm$^{-3}$), a moderate magnetic field strength (B
$\approx$ 140 G), and an energy distribution of electrons following a power-law
($d N / d E \propto E^{- \delta}$) with $\delta$ fixed at 1.36. These radiation
belts would need to be present in both components and also be viewed
equatorially.
