Potentially semi-stable deformations of specified Hodge-Tate type and Galois type

Let $k$ be a perfect field of characteristic $p>2$, and let $K$ be a finite totally ramified extension of $W(k)[\frac{1}{p}]$. We prove that the locus of potentially semi-stable $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{K}\mathrm{/}K)$-representations of a given Hodge-Tate type and Galois type is a closed subspace of the universal deformation ring, generalizing the result of Kisin (2007) where $k$ is assumed to be finite.