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 ~ 105 - 107 K. However, direct observation of this so-called warm-hot intergalactic medium (WHIM) has been elusive. In theΛcold dark matter 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 H I Lyα absorbers (BLAs) potentially produced by WHIM in inter-cluster filaments. We targeted a single QSO, namely Q1410, whose sightline intersects seven independent inter-cluster axes at impact parameters < 3Mpc (comoving), and found a tentative excess of a factor of ~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 seven BLAs in inter-cluster axes, we found three without any galaxy counterpart to stringent luminosity limits (~4 × 108 L⊙ ~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 Ωbarfil = 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.