  Modern analyses of structure formation predict a universe tangled in a
'cosmic web' of dark matter and diffuse baryons. These theories further predict
that at low-z, a significant fraction of the baryons will be shock-heated to $T
\sim 10^{5}-10^{7}$K yielding a warm-hot intergalactic medium (WHIM), but whose
actual existence has eluded a firm observational confirmation. We present a
novel experiment to detect the WHIM, by targeting the putative filaments
connecting galaxy clusters. We use HST/COS to observe a remarkable QSO
sightline that passes within $\Delta d = 3$ Mpc from the 7 inter-cluster axes
connecting 7 independent cluster-pairs at redshifts $0.1 \le z \le 0.5$. We
find tentative excesses of total HI, narrow HI (NLA; Doppler parameters $b<50$
km/s), broad HI (BLA; $b \ge 50$ km/s) and OVI absorption lines within
rest-frame velocities of $\Delta v \lesssim 1000$ km/s from the cluster-pairs
redshifts, corresponding to $\sim 2$, $\sim 1.7$, $\sim 6$ and $\sim 4$ times
their field expectations, respectively. Although the excess of OVI likely comes
from gas close to individual galaxies, we conclude that most of the excesses of
NLAs and BLAs are truly intergalactic. We find that the covering fractions,
$f_c$, of BLAs close to cluster-pairs are $\sim 4-7$ times higher than the
random expectation (at the $\sim 2 \sigma$ c.l.), whereas the $f_c$ of NLAs and
OVI are not significantly enhanced. We argue that a larger relative excess of
BLAs compared to those of NLAs close to cluster-pairs may be a signature of the
WHIM in inter-cluster filaments. By extending the present analysis to tens of
sightlines our experiment offers a promising route to detect the WHIM.
