Do precision electroweak constraints guarantee e+e− collider discovery of at least one Higgs boson of a two-Higgs-doublet model?

We consider a CP-conserving two-Higgs-doublet type II model with a light scalar or pseudoscalar neutral Higgs boson (h=h0 or h=A0) that has no ZZ/WW coupling and, thus, cannot be detected in e+e−→Zh (Higgs-strahlung) or νν̄h (via WW fusion). Despite sum rules which ensure that the light h must have significant tt̄ or bb̄ coupling, for a wedge of moderate tanβ, that becomes increasingly large as mh increases, the h can also escape discovery in both bb̄h and tt̄h production at a s=500–800 GeV e+e− collider (for expected luminosities). If the other Higgs bosons happen to be too heavy to be produced, then no Higgs boson would be detected. We demonstrate that, despite such high masses for the other Higgs bosons, only the low-tanβ portion of the no-discovery wedges in [mh,tanβ] parameter space can be excluded due to failure to fit precision electroweak observables. In the tanβ≳1 regions of the no-discovery wedges, we find that the 2HDM fit to precision electroweak observables has small Δχ2 relative to the best minimal one-doublet SM fit.