Suppose we have a de Rham Galois representation $G_K\rightarrow GL(V)$ for some $p$-adic field $K$ and some finite dimensional vector space $V$ over $\mathbf{Q}_p$. Then it is a theorem that there is some finite extension $L/K$ such that $D_{dR}^L(V)\cong L\otimes_{L_0}D_{st}^L(V)$ (this is actually a corollary of the theorem that de Rham implies potentially semi-stable).

My question is whether I can still find such an extension $L/K$ without the assumption that the representation is de Rham. In other words, is there always some finite extension $L/K$ such that $dim_L D_{dR}^L(V)=dim_{L_0}D_{st}^L(V)$? I can't imagine that this is true, so what I'm really asking for is a counterexample.