  Let K/Q_p be unramified. Inside the Emerton-Gee stack X_2, one can consider
the locus of two-dimensional mod p representations of the absolute Galois group
of K having a crystalline lift with specified Hodge-Tate weights. We study the
case where the Hodge-Tate weights are irregular, which is an analogue for
Galois representations of the partial weight one condition for Hilbert modular
forms. We prove that if the gap between each pair of weights is bounded by p
(the irregular analogue of a Serre weight), then this locus is irreducible. We
also establish various inclusion relations between these loci.
