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)[1p]. We prove that the locus of potentially semi-stable Gal(K¯/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) [Kis07] where k is assumed to be finite.