  Cosmological simulations predict that a significant fraction of the low-$z$
baryon budget resides in large-scale filaments in the form of a diffuse plasma
at temperatures $T \sim 10^{5} - 10^{7}$ K. However, direct observation of this
so-called warm-hot intergalactic medium (WHIM) has been elusive. In the
$\Lambda$CDM paradigm, galaxy clusters correspond to the nodes of the cosmic
web at the intersection of several large-scale filamentary threads. In previous
work, we used HST/COS data to conduct the first survey of broad HI Ly$\alpha$
absorbers (BLAs) potentially produced by WHIM in inter-cluster filaments. We
targeted a single QSO, namely Q1410, whose sight-line intersects $7$
independent inter-cluster axes at impact parameters $<3$ Mpc (co-moving), and
found a tentative excess of a factor of ${\sim}4$ with respect to the field.
Here, we further investigate the origin of these BLAs by performing a blind
galaxy survey within the Q1410 field using VLT/MUSE. We identified $77$ sources
and obtained the redshifts for $52$ of them. Out of the total sample of $7$
BLAs in inter-cluster axes, we found $3$ without any galaxy counterpart to
stringent luminosity limits ($\sim 4 \times 10^{8}$ L$_{\odot}$ ${\sim} 0.01$
L$_{*}$), providing further evidence that these BLAs may represent genuine WHIM
detections. We combined this sample with other suitable BLAs from the
literature and inferred the corresponding baryon mean density for these
filaments in the range $\Omega^{\rm fil}_{\rm bar}= 0.02-0.04$. Our rough
estimates are consistent with the predictions from numerical simulations but
still subject to large systematic uncertainties, mostly from the adopted
geometry, ionization corrections and density profile.
