  Multi-Higgs boson interactions with massive gauge bosons are known to be
tell-tale probes of the vacuum manifold of electroweak symmetry breaking.
Phenomenologically, a precise determination of these parameters is hampered
through increasingly rare processes at the presently available energy frontier
provided by the Large Hadron Collider. Contact interactions of three Higgs
bosons with the $W$ and $Z$ bosons seem currently well out of experimental
reach due to an irrelevant SM production cross section. From a theoretical
perspective, in perturbative extensions of the SM such interactions are
suppressed by weak loops and further diluted in a priori sensitive processes
like weak boson fusion (WBF) when they admit a dimension-six Standard Model
Effective Field Theory description. In this work, we identify scenarios that
can indeed lead to large, and perhaps even observable modifications of WBF
triple Higgs production most directly parametrised by Higgs Effective Field
Theory. We critically analyse these enhancements at the LHC and future
colliders from the perspective of unitarity and demonstrate the radiative
stability of such analyses under QCD corrections at hadron colliders. In
passing, we also comment on the expected sensitivity at non-hadron colliders.
