  We investigate the possible presence of neutral hydrogen (HI) in
intergalactic filaments at very low redshift ($z\sim 0.08$), by stacking a set
of 274,712 2dFGRS galaxy pairs over 21-cm maps obtained with dedicated
observations conducted with the Parkes radio telescope, over a total sky area
of approximately 1,300 square degrees covering two patches in the northern and
in the southern Galactic hemispheres. The stacking is performed by combining
local maps in which each pair is brought to a common reference frame; the
resulting signal from the edge galaxies is then removed to extract the filament
residual emission. We repeat the analysis on maps cleaned removing either 10 or
20 foreground modes in a principal component analysis. Our study does not
reveal any clear HI excess in the considered filaments in either case; we
determine upper limits on the total filament HI brightness temperature at
$T_{\rm b} \lesssim 10.3 \,\mu\text{K}$ for the 10-mode and at $T_{\rm b}
\lesssim 4.8 \,\mu\text{K}$ for the 20-mode removed maps at the 95% confidence
level. These estimates translate into upper limits for the local filament HI
density parameter, $\Omega_{\rm HI}^{\rm (f)} \lesssim 7.0\times10^{-5}$ and
$\Omega_{\rm HI}^{\rm (f)} \lesssim 3.2\times10^{-5}$ respectively, and for the
HI column density, $N_{\rm HI} \lesssim 4.6\times10^{15}\,\text{cm}^{-2}$ and
$N_{\rm HI} \lesssim 2.1\times10^{15}\,\text{cm}^{-2}$ respectively. These
column density constraints are consistent with previous detections of HI in the
warm-hot intergalactic medium obtained observing broad Ly $\alpha$ absorption
systems. The present work shows for the first time how such constraints can be
achieved using the stacking of galaxy pairs on 21-cm maps.
