  Previous work of Kisin and Gee proves potential diagonalisability of two
dimensional Barsotti-Tate representations of the Galois group of a finite
extension $K/\mathbb{Q}_p$. In this paper we build upon their work by relaxing
the Barsotti-Tate condition to one we call pseudo-Barsotti-Tate (which means
that for certain embeddings $\kappa:K \rightarrow \overline{\mathbb{Q}}_p$ we
allow the $\kappa$-Hodge-Tate weights to be contained in $[0,p]$ rather than
$[0,1]$).
