For certain classes of Beyond the Standard Model theories, including composite Higgs models, the coupling of the Higgs to gauge bosons can be different from the Standard Model one. In this case, the multi-boson production via vector boson fusion (VBF) can be hugely enhanced in comparison to the SM production one due to the lack of cancellation in longitudinal vector boson scattering. Among these processes, triple Higgs boson production in VBF plays a special role - its enhancement is especially spectacular due to the absence of background from transversely polarised vector bosons in the final state. While the rates from $ppto jjhhh$ production in vector boson fusion are too low at the LHC and even at future 33 TeV $pp$ colliders, we have found that the 100 TeV $pp$ future circular collider (FCC) has the unique opportunity to probe the $hVV$ coupling far beyond the LHC sensitivity. We have evaluated the $ppto jjhhh$ rates as a function of deviation from the $hVV$ coupling and have found that the background is much smaller than the signal for observable signal rates. We also found that the 100 TeV $pp$ FCC can probe the $hVV$ coupling up to the permille level, which is far beyond the LHC reach. These results highlight a special role of the $hhh$ VBF production and stress once more the importance of the 100 TeV $pp$ FCC.
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