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Constraints on quartic interactions of the Higgs boson with gauge bosons have been obtained by the experimental LHC collaborations focussing on the so-called $κ$ framework of flat rescalings of SM-like interactions in weak boson fusion (WBF) Higgs pair production. While such approaches are admissible to obtain a qualitative picture of consistency with the SM when the statistical yield is low, once more statistics become available a more theoretically consistent framework of limit setting is desirable. Reviewing the constraints provided at the Large Hadron Collider, we first show that these limits are robust when considered in a leading order context. Turning to radiative corrections, we demonstrate the limitations of this approach in the SM, and by adopting Higgs effective field theory techniques, we clarify the sensitivity from single Higgs measurements to rescalings of quartic Higgs-gauge couplings. We then discuss avenues for sensitivity improvements of WBF analyses employing Graph Neural Networks to combat the large contributing backgrounds.