Vanishing conjecture for high tensor-power Selmer groups

Let $X/F$ be a smooth curve, let $\bar X$ be its base change to an algebraic closure, and set

$$V_n=H^1_{\mathrm{et}}(\bar X,\mathbb{Q}_p)^{\otimes n}(1).$$

Let $T$ be the finite set of places under consideration and define

$$Sel^0_T(V_n)=\ker\left(H^1(G_T,V_n)\longrightarrow\bigoplus_{w\in T}H^1(G_w,V_n)\right).$$

Vanishing conjecture for high tensor powers. One has $Sel^0_T(V_n)=0$ for $n\gg0$. This conjecture concerns the disappearance of globally unramified Selmer classes in sufficiently high tensor powers of the étale cohomology representation. The source gives no resolution and uses it in relating the non-abelian Selmer dimension problem to standard Selmer-type conjectures.