Density of potentially crystalline representations of fixed weight

Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $\bar{\rho}$ be a
continuous, absolutely irreducible representation of its absolute Galois
group with values in a finite field of characteristic p. We prove that the
Galois representations that become crystalline of a fixed regular weight
after an abelian extension are Zariski-dense in the generic fiber of the
universal deformation ring of $\bar{\rho}$. In fact we deduce this from a
similar density result for the space of trianguline representations. This
uses an embedding of eigenvarieties for unitary groups into the spaces of
trianguline representations as well as the corresponding density claim for
eigenvarieties as a global input.