  We consider the decay of the $Z$ boson into three pseudoscalar bosons in a
general two-Higgs-doublet model. Assuming $m_A$ to be very small, and that of
the two physical neutral scalar bosons $h_1$ and $h_2$, $A$ only couples to $Z$
through $h_1$, we find the $Z \to A A A$ branching fraction to be negligible
for moderate values of $\tan \beta \equiv v_2/v_1$, if there is no $\lambda_5
(\Phi_1^\dagger \Phi_2)^2 + h.c.$ term in the Higgs potential; otherwise there
is no absolute bound but very large quartic couplings (beyond the validity of
perturbation theory) are needed for it to be observable.
