Voisin's integral Hodge/Tate conjecture for one cycles on rationally connected varieties

Let $X$ be a smooth projective separably rationally connected variety of dimension $d$ over an algebraically closed field. The groups

$$H^{2d-2}_{\operatorname{\acute{e}t}}(X, \mathbb{Z}_l(d-1))$$

and, when $X$ is defined over the complex numbers,

$$H^{2d-2}_{\operatorname{sing}}(X, \mathbb{Z})$$

are algebraic. Voisin's integral Hodge/Tate conjecture. Every class in these integral cohomology groups is generated by classes of algebraic cycles. The usual Tate/Hodge conjecture gives algebraicity only for the relevant Galois-invariant or $(p,p)$ classes; for separably rationally connected varieties, the corresponding rational cohomology is already algebraic by a decomposition of the diagonal. The integral assertion is known in characteristic zero for threefolds and smooth Fano fourfolds, but remains open in general.