A search for a Higgs boson with suppressed couplings to fermions, $h_f$, assumed to be the neutral, lower-mass partner of the Higgs boson discovered at the Large Hadron Collider, is reported. Such a Higgs boson could exist in extensions of the standard model with two Higgs doublets, and could be produced via $pbar{p} to H^pm h_f to W^* h_f h_f to 4gamma + X$, where $H^pm$ is a charged Higgs boson. This analysis uses all events with at least three photons in the final state from proton-antiproton collisions at a center-of-mass energy of 1.96~TeV collected by the Collider Detector at Fermilab, corresponding to an integrated luminosity of 9.2~${rm fb}^{-1}$. No evidence of a signal is observed in the data. Values of Higgs-boson masses between 10 and 100 GeV/$c^2$ are excluded at 95% Bayesian credibility.
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