  A broad class of models in which electroweak symmetry breaking originates
from dynamics in a singlet dark sector, and is transferred to the Standard
Model via the Higgs portal, predicts in general strongly suppressed Higgs boson
mixing with a singlet scalar. In this work we point out that at present this
class of models allows for the second phenomenologically acceptable solution
with almost maximal mixing between the Higgs and the scalar singlet. This
scenario predicts an almost degenerate twin peak Higgs signal which is
presently indistinguishable from a single peak, due to the limited LHC mass
resolution. Because of that, the LHC experiments measure inclusive Higgs rates
that all must exactly agree with Standard Model predictions due to sum rules.
We show that if the dark sector and Standard Model communicate only via the
singlet messenger scalar that mixes with the Higgs, the spin independent direct
detection cross section of dark matter is suppressed by the scalar mass
degeneracy, explaining its non-observation so far.
