  Recent atmospheric models for brown dwarfs suggest that the existence of
clouds in substellar objects is not needed to reproduce their spectra, nor
their rotationally-induced photometric variability, believed to be due to the
heterogeneous cloud coverage of brown dwarf atmospheres. Cloud-free atmospheric
models also predict that their flux should not be polarized, as polarization is
produced by the light-scattering of particles in the inhomogeneous cloud layers
of brown dwarf atmospheres. To shed light on this dichotomy, we monitored the
linear polarization and photometric variability of the most variable brown
dwarf, 2MASS J21392676+0220226. We used FORS2 at the UT1 telescope to monitor
the object in the z-band for six hours, split on two consecutive nights,
covering one-third of its rotation period. We obtained the Stokes parameters,
and we derived its time-resolved linear polarization, for which we did not find
significant linear polarization (P = 0.14+\-0.07 %). We modeled the linear
polarimetric signal expected assuming a map with one or two spot-like features
and two bands using a polarization-enabled radiative-transfer code. We obtained
values compatible with the time-resolved polarimetry obtained for 2MASS
J21392676+0220226. The lack of significant polarization might be due to
photometric variability produced mostly by banded structures or small-scale
vortices, which cancel out the polarimetric signal from different regions of
the dwarf's disk. Alternatively, the lack of clouds in 2MASS J21392676+0220226
would also explain the lack of polarization. Further linear polarimetric
monitoring of 2MASS J21392676+0220226, during at least one full rotational
period, would help to confirm or discard the existence of clouds in its
atmosphere.
