  The electroweak symmetry-breaking sector is one of the most promising and
uncharted parts of the Standard Model; but it seems likely that new electroweak
physics may be out of reach of the present accelerator effort and the hope is
to observe small deviations from the SM. Given that, Effective Field Theory
becomes the logic method to use, and SMEFT has become the standard. However,
the most general theory with the known particle content is HEFT, and whether
SMEFT suffices should be investigated in future experimental efforts. Building
on investigations by other groups that established geometric criteria to
distinguish SMEFT from HEFT (useful for theorists examining specific beyond-SM
completions), we seek more phenomenological understanding and present an
analogous discussion aimed at a broader audience.
  We discuss various aspects of (multi-) Higgs boson production from
longitudinal electroweak gauge bosons $W_LW_L\to n\times h$ in the TeV region
as the necessary information to characterise the Flare function,
$\mathcal{F}(h)$, that determines whether SMEFT or HEFT is needed. We also
present tree-level amplitudes including contact and exchange channels, as well
as a short discussion on accessing $\mathcal{F}(h)$ from the statistical limit
of many bosons. We also discuss the status of the coefficients of its series
expansion, its validity, whether its complex-$h$ extension can be used to
predict or not a tell-tale zero, and how they relate to the dimension-6 and -8
SMEFT operators in the electroweak sector. We derive a set of new correlations
among BSM corrections to the HEFT coefficients that help decide, from
experimental data, whether we have a viable SMEFT. This analysis can be useful
for machines beyond the LHC that could address the challenging final state with
several Higgs bosons.
