I may not be aware of the very latest developments, but as far as I see, after a little BICEP2 hickup, the Starobinsky model of cosmic inflation (or the class of these models) is rather preferred by observational data. Is that (still) right?
It's curious, because several arguments (as behind the above link) seem to suggest that Starobinsky inflation goes rather well along with supergravity and low energy supersymmetry breaking.
This here is a vague question, just out of interest, sorry. I am wondering if any expert seeing this here might enjoy providing an update on the status of the models, particularly with respect to embedding into supergravity and superstring models.