The Witten-Sakai-Sugimoto model, a top-down gauge/gravity model of large-$N_c$ low-energy QCD based on type-IIA string theory and $N_c$ D4 branes with chiral quarks added by probe D8 branes, involves only one dimensionless parameter, the 't Hooft coupling at its cutoff (Kaluza-Klein) scale. Although this cutoff scale is around 1 GeV, the model has turned out to be surprisingly predictive also quantitatively, reproducing masses of vector and axial vector mesons, their decay rates, as well as the anomalous mass of the $\eta'$ meson, all within 10-30% errors. Using it as a guide for glueball signatures, we have argued that it indicates that the meson $f_0(1710)$ may be a nearly pure glueball, with rather specific predictions for the still-to-be-measured $4\pi$ and $\eta\eta'$ decays. Here we present our new predictions for the decay pattern of the (very narrow) pseudoscalar glueball, which is closely related to the U(1)$_A$ problem that the model is in fact handling correctly.