Let $L/\mathbb{Q}$ be a finite extension and let $V$ be a one-dimensional $L$-linear representation of $G_{\mathbb{Q}}$ which is given by $\chi\rho^*\kappa^n_{cyc}:G_{\mathbb{Q}}\rightarrow L^\times$, where $\rho$ is a Dirichlet character mod $p$, $\chi$ is a Dirichlet character mod $d$ for some $d$ prime to $p$ (both viewed as galois characters) and $\kappa_{cyc}$ is the cyclotomic character for the prime $p$. In particular the restriction of $V$ to $G_{\mathbb{Q}_p}$ is de Rham. Further we assume $n\leq0$ and $\chi\rho^*(-1)\cdot(-1)^n=-1$. To be precise, this representation comes from the Artin motive corresponding to $\chi$ twisted by $\rho^*$ and the Tate motive $\mathbb{Q}(n)$.
Let $S=\{\text{primes dividing } $d$,p,\infty\}$ as a subset of the places of $\mathbb{Q}$. I want to show (if true) that
the Bloch-Kato-Selmer group
$$
H^1_f(\mathbb{Q},V)=\ker\left(H^1(G_{\mathbb{Q},S},V)\longrightarrow\bigoplus_{v\in S}\frac{H^1(\mathbb{Q}_v,V)}{H_f^1(\mathbb{Q}_v,V)}\right),
$$
is zero. Here the $H_f^1(\mathbb{Q}_v,V)$ are "local conditions", namely
$$
H_f^1(\mathbb{Q}_v,V)=\begin{cases}\ker(H^1(\mathbb{Q}_l,V)\overset{res}{\longrightarrow}H^1(I_l,V)),& v=l\neq p\\
\ker(H^1(\mathbb{Q}_p,V)\longrightarrow H^1(\mathbb{Q}_p,B_{cris}\otimes_{\mathbb{Q}_p},V)),&v=p\\
H^1(\mathbb{R},V),&v=\infty,
\end{cases}
$$
where $I_l\subseteq G_{\mathbb{Q}_l}$ is the inertia subgroup.
One can show relatively easy that in this case one has $H^1_f(\mathbb{Q}_v,V)=0$ for all $v\in S$ and therefore it reduces to the question wether the usual localization map
$$
H^1(G_{\mathbb{Q},S},V)\longrightarrow\bigoplus_{v\in S} H^1(\mathbb{Q}_v,V)
$$
is injective. To go further, I assume one has to use that $H^1(G,V)=\projlim_{n\in\mathbb{N}}H^1(G,T/p^nT)\otimes_{\mathcal{O}_L}L$, where $T\subseteq V$ is a $G$-stable $\mathcal{O}_L$-lattice and $G\in\{G_{\mathbb{Q},S},G_{\mathbb{Q}_v}\}$, and then use Pitou-Tate duality. In https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/NSW2.3.pdf, §9.1 they discuss some cases where the localization map is zero, but it is mainly about finite simple $G$-modules and I don't see how I can apply this here. One can also show that $H^1_f(\mathbb{Q},V)=0$ if and only if
$$
H^1_f(\mathbb{Q},V/T)=\ker\left(H^1(G_{\mathbb{Q},S},V/T)\longrightarrow\bigoplus_{v\in S}\frac{H^1(\mathbb{Q}_v,V/T)}{H_f^1(\mathbb{Q}_v,V/T)}\right),
$$
is finite. But I don't know how to prove this either. It would be great if someone could help.