The integral Tate conjecture for separably rationally connected varieties in codimension one

Let $X$ be a smooth projective variety over a finite field, and let $d=\dim X$. Say that $X$ is separably rationally connected in codimension $1$ if it admits a morphism $f:\mathbb{P}^1\to X$ such that, in the splitting $f^*T_X=\bigoplus_i\mathcal{O}(a_i)$, all $a_i\geq 0$ and all but one are strictly positive. Integral Tate conjecture. If $X$ is separably rationally connected in codimension $1$, then

$$CH_1(X)\otimes\mathbb{Z}_\ell\longrightarrow H^{2d-2}_{\text{ét}}(X,\mathbb{Z}_\ell(d-1))$$

is surjective. This is motivated by the failure of the integral cycle class map to be surjective in general, and the source presents it as a conjectural consequence of the paper's results and earlier work.