Splitting families in Galois cohomology

Let $k$ be a field, with absolute Galois group $\mathrm{\Gamma }$. Let $A\mathrm{/}k$ be a finite 'etale group scheme of multiplicative type, i.e. a discrete $\mathrm{\Gamma }$-module. Let $n\ge 2$ be an integer, and let $x\in H^n(k,A)$ be a cohomology class. We show that there exists a countable set $I$, and a familiy $(X_i{)}_{i\in I}$ of (smooth, geometrically integral) $k$-varieties, such that the following holds. For any field extension $l\mathrm{/}k$, the restriction of $x$ vanishes in $H^n(l,A)$ if and only if (at least) one of the $X_i$’s has an $l$-point. We moreover show that the $X_i$’s can be made into an ind-variety. In the case $n=2$, we note that one variety is enough.