Let $k$ be a $p$-adic field and $G_k$ its absolute Galois group. Let $B_\text{dR}$ be the de Rham period ring with the usual filtration given by powers of $t$. For $i < j$ integers we have an exact sequence $$ 0 \rightarrow t^{i+1}B^+_\text{dR}/t^{j+1}B^+_\text{dR} \rightarrow t^{i}B^+_\text{dR}/t^{j+1}B^+_\text{dR} \rightarrow t^iB^+_\text{dR}/t^{i+1}B^+_\text{dR} \rightarrow 0.$$

Taking $G_k$-invariants gives us again a short exact sequence (we can use $t^iB^+_\text{dR}/t^{i+1}B^+_\text{dR} \cong \hat{\overline{k}}(i)$ to calculate this).

Now let $V$ be a $p$-adic representation of $G_k$. Can we show that $$0 \rightarrow (t^{i+1}B^+_\text{dR}/t^{j+1}B^+_\text{dR} \otimes V)^{G_k}\rightarrow (t^{i}B^+_\text{dR}/t^{j+1}B^+_\text{dR} \otimes V)^{G_k}\rightarrow (t^iB^+_\text{dR}/t^{i+1}B^+_\text{dR} \otimes V)^{G_k} \rightarrow 0$$ is also exact?

Is it true if we assume some nice properties of $V$, like Hodge–Tate, de Rham etc.?