There has been great interest in a model with three Higgs doublets in which
fermions with a particular charge couple to a single and distinct Higgs field.
We study the phenomenological differences between the two common incarnations
of this so-called Type-Z 3HDM. We point out that the differences between the
two models arise from the scalar potential only. Thus we focus on observables
that involve the scalar self-couplings. We find it difficult to uncover
features that can uniquely set apart the Z3​ variant of the model. However,
by studying the dependence of the trilinear Higgs couplings on the nonstandard
masses, we have been able to isolate some of the exclusive indicators for the
Z2​×Z2​ version of the Type-Z 3HDM. This highlights the importance of
precision measurements of the trilinear Higgs couplings.Comment: 15 pages, 4 captioned figure