  Indirect constraints on the total Higgs width $\Gamma_h$ 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 $\Gamma_h$ obtained in this fashion are not
reliable unless we make model-specific assumptions, which cannot be justified
at the current stage of the LHC programme. Relaxing the $\Gamma_h$
interpretation, we discuss the merits of high invariant mass cross section
measurements in the context of Higgs CP 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.
