Indirect constraints on the total Higgs width ${\mathrm{\Gamma }}_h$ from correlating Higgs signal strengths with cross-section measurements in the off-shell region for $p(g)p(g)\to 4\ell$ production have received considerable attention recently, and the CMS Collaboration have published a first measurement. We revisit this analysis from a new physics and unitarity constraints perspective and conclude that limits on ${\mathrm{\Gamma }}_h$ obtained in this fashion are not reliable unless we make model-specific assumptions, which cannot be justified at the current stage of the LHC program. Relaxing the ${\mathrm{\Gamma }}_h$ interpretation, we discuss the merits of high invariant mass cross-section measurements in the context of Higgs $\mathcal{C}\mathcal{P}$ analyses, higher-dimensional operator testing, and resolved new physics in the light of electroweak precision constraints beyond effective theory limitations. Furthermore, we show that a rather model-independent LHC constraint can be obtained from adapting the $gg\to 4\ell$ analysis to the weak boson fusion channels at lower statistical yield.

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