  We show that the theoretical machinery developed for the Kardar-Parisi-Zhang
(KPZ) class in low dimensions are obeyed by the restricted solid-on-solid
(RSOS) model for substrates with dimensions up to $d=6$. Analyzing different
restriction conditions, we show that height distributions of the interface are
universal for all investigated dimensions. It means that fluctuations are not
negligible and, consequently, the system is still below the upper critical
dimension at $d=6$. The extrapolation of the data to dimensions $d\ge7$
predicts that the upper critical dimension of the KPZ class is infinite.
