The localisation injectivity and vanishing conjecture for symmetric powers

Let $V$ be the Galois representation used in the paper, let $G_T$ be the global Galois group outside the specified set $T$ of places, and let $G_v$ denote the corresponding local Galois groups. For every integer $i$ with $0\leq i\leq n$, consider the localisation map

$$\sum \operatorname{loc}_v H^1(G_T,\operatorname{Sym}^i(V)(1))\longrightarrow\bigoplus_{v\in T}H^1(G_v,\operatorname{Sym}^i(V)(1)).$$

Localisation injectivity and vanishing conjecture. For all $0\leq i\leq n$, this map is injective, and

$$H^1_f(G_T,\operatorname{Sym}^i(V)(-i))=0.$$

The conjecture is introduced as an assumption needed to describe the localisation map from the Selmer variety. The supplied text gives no resolution or further evidence, so its status remains open here.