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In a recently proposed neutrinophilic two-Higgs-doublet model, the low-energy (sub-MeV) effective theory consists of a real scalar with a vacuum expectation value of O(0.1)??eV and three Dirac neutrinos. Other models could lead to the same low-energy theory. In this Brief Report, we study constraints on the parameter space of the model, including vacuum stability, unitarity, perturbativity, and the effects on the invisible Z width. Interestingly, we find that all neutrinos become massless at temperatures above approximately 1000 K, but can find no phenomenological effects of this finding. The most direct test of the model is that it predicts that, in a galactic supernova, the energy distributions of the electron, muon, and tau neutrinos will be Fermi-Dirac with identical temperatures, unlike the conventional distributions. [ABSTRACT FROM AUTHOR]