  We prove that all mod $p$ Hilbert modular forms arise via multiplication by
generalized partial Hasse invariants from forms whose weight falls within a
certain minimal cone. This answers a question posed by Andreatta and Goren, and
generalizes our previous results which treated the case where $p$ is unramified
in the totally real field. Whereas our previous work made use of deep
Jacquet-Langlands type results on the Goren-Oort stratification (not yet
available when $p$ is ramified), here we instead use properties of the
stratification at Iwahori level which are more readily generalizable to other
Shimura varieties.
