The global-to-local injectivity conjecture for H²

Let $K$ be a number field, let $X/K$ be smooth and projective, let $S$ be a finite set of places as in the source, and let $m,s$ be integers. Global-to-local injectivity conjecture. If $m-2s\neq -1$, then the map

$$H^2(G_S,H^m(\overline X,\mathbb Q_\ell(s)))\longrightarrow\bigoplus_{v\in S}H^2(K_v,H^m(\overline X,\mathbb Q_\ell(s)))$$

is injective. The source presents this as a conjectural input implying the preceding global vanishing statement; no resolution is supplied.