  Strong accretion shocks are expected to illuminate the warm-hot
inter-galactic medium encompassed by the filaments of the cosmic web, through
synchrotron radio emission. Given their high sensitivity, low-frequency large
radio facilities may already be able to detect signatures of this extended
radio emission from the region in between two close and massive galaxy
clusters. In this work we exploit the non-detection of such diffuse emission by
deep observations of two pairs of relatively close ($\simeq 10$ Mpc) and
massive ($M_{500}\geq 10^{14}M_\odot$) galaxy clusters using the LOw-Frequency
ARray (LOFAR). By combining the results from the two putative inter-cluster
filaments, we derive new independent constraints on the median strength of
inter-galactic magnetic fields: $B_{\rm Mpc}< 2.5\times 10^2\,\rm nG\,(95\%\,
\rm CL)$. Based on cosmological simulations and assuming a primordial origin of
the B-fields, these estimates can be used to limit the amplitude of primordial
seed magnetic fields: $B_0\leq10\,\rm nG$. We advise the observation of similar
cluster pairs as a powerful tool to set tight constraints on the amplitude of
extragalactic magnetic fields.
