Potential diagonalisability of pseudo-Barsotti-Tate representations

Previous work of Kisin and Gee proves potential diagonalisability of two dimensional Barsotti-Tate representations of the Galois group of a finite extension $K\mathrm{/}{\mathbb{Q}}_p$. 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 $\kappa :K\to {\bar{\mathbb{Q}}}_p$ we allow the $\kappa$-Hodge-Tate weights to be contained in $[0,p]$ rather than $[0,1]$).