Bloch–Kato–Selmer group of a one-dimensional representation

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 $chirho^*kappa^n_{text{cyc}}:G_{mathbb{Q}}rightarrow L^times$, 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 $kappa_{text{cyc}}$ 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 $nleq0$ and $chirho^*(-1)cdot(-1)^n=-1$. 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
$$
H^1_f(mathbb{Q},V)=kerleft(H^1(G_{mathbb{Q},S},V)longrightarrowbigoplus_{vin S}frac{H^1(mathbb{Q}_v,V)}{H_text f^1(mathbb{Q}_v,V)}right),
$$
is zero. Here the $H_text f^1(mathbb{Q}_v,V)$ are "local conditions", namely
$$
H_text f^1(mathbb{Q}_v,V)=begin{cases}ker(H^1(mathbb{Q}_l,V)overset{operatorname{res}}{longrightarrow}H^1(I_l,V)),& v=lneq p\
ker(H^1(mathbb{Q}_p,V)longrightarrow H^1(mathbb{Q}_p,B_{text{cris}}otimes_{mathbb{Q}_p},V)),&v=p\
H^1(mathbb{R},V),&v=infty,
end{cases}
$$
where $I_lsubseteq G_{mathbb{Q}_l}$ is the inertia subgroup.
One can show relatively easily that in this case one has $H^1_text f(mathbb{Q}_v,V)=0$ for all $vin S$ and therefore it reduces to the question of whether the usual localization map
$$
H^1(G_{mathbb{Q},S},V)longrightarrowbigoplus_{vin S} H^1(mathbb{Q}_v,V)
$$
is injective. To go further, I assume one has to use that $H^1(G,V)=projlim_{ninmathbb{N}}H^1(G,T/p^nT)otimes_{mathcal{O}_L}L$, where $Tsubseteq V$ is a $G$-stable $mathcal{O}_L$-lattice and $Gin{G_{mathbb{Q},S},G_{mathbb{Q}_v}}$, and then use Pitou–Tate duality. In Neukirch, Schmidt, and Wingberg – Cohomology of number fields, §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 $H_text f^1(mathbb{Q},V)=0$ if and only if
$$
H_text f^1(mathbb{Q},V/T)=kerleft(H^1(G_{mathbb{Q},S},V/T)longrightarrowbigoplus_{vin S}frac{H^1(mathbb{Q}_v,V/T)}{H_text f^1(mathbb{Q}_v,V/T)}right)
$$
is finite. But I don’t know how to prove this either. It would be great if someone could help.