Positive-characteristic structure conjecture for Suslin homology

Let $\Bbbk$ be an algebraically closed field of positive characteristic $p$, and let $X$ be a scheme separated of finite type over $\Bbbk$. Define $H_i^S(X,\mathbb{Z}[1/p])$, and let $b_{i,\ell}(X)=\operatorname{rank}(H^i_{\operatorname{\acute{e}t}}(X,\mathbb{Z}_{\ell}))$.

Positive-characteristic structure conjecture. There are a uniquely divisible group $V$, integers $s_{i,\ell}$ and $r_i$, and a torsion group $T=\bigoplus_{\ell\neq p}H^{i+1}_{\operatorname{\acute{e}t}}(X,\mathbb{Z}_{\ell})_{\mathrm{tor}}$ such that

$$H_i^S(X,\mathbb{Z}[1/p])\cong V\oplus \bigoplus_{\ell\neq p}(\mathbb{Q}_p/\mathbb{Z}_p)^{s_{i,\ell}}\oplus \mathbb{Z}[1/p]^{\oplus r_i}\oplus T,$$

and

$$s_{i-1,\ell}+r_i=b_{i,\ell}(X)=\operatorname{rank}(H^i_{\operatorname{\acute{e}t}}(X,\mathbb{Z}_{\ell})).$$

This conjecture implies $H_S^i(X,\mathbb{Z}/n)\cong H_{\operatorname{\acute{e}t}}^i(X,\mathbb{Z}/n)$, a result established in the cited work. The source does not provide evidence that the displayed structural conjecture itself has been resolved.