  We present a search for new particles in an extension to the standard model
that includes a heavy Higgs boson ($H^0$), a lighter charged Higgs boson
($H^\pm$), and an even-lighter Higgs boson $h^0$, with decays leading to a
$W$-boson pair and a bottom-antibottom quark pair in the final state. We use
events with exactly one lepton, missing transverse momentum, and at least four
jets in data corresponding to an integrated luminosity of 8.7 fb$^{-1}$
collected by the CDF II detector in proton-antiproton collisions at
$\sqrt{s}=1.96$ TeV. We find the data to be consistent with standard model
predictions and report the results in terms of a simplified Higgs-cascade-decay
model, setting 95% confidence level upper limits on the product of
cross-section and branching fraction from 1.3 pb to 15 fb as a function of
$H^0$ and $H^\pm$ masses for $m_h^{0}=126$ GeV$/c^2$.
