  The previously proposed class of phenomenological inflationary models in
which the assumption of inflaton slow-roll is replaced by the more general,
constant-roll condition is compared with the most recent cosmological
observational data, mainly the Planck ones. Models in this two-parametric class
which remain viable appear to be close to the slow-roll ones, and their
inflaton potentials are close to (but still different from) that of the natural
inflation model. Permitted regions for the two model parameters are presented.
