The weak bi-anabelian m-step solvable Grothendieck conjecture in positive characteristic

Let $k$ be a finitely generated field of characteristic $p>0$ with absolute Galois group $G_k$, let $m$ be a non-negative integer, and let $U_i$ be hyperbolic curves over $k$ for $i=1,2$. Let $U_i(n)$ denote the $n$-th relative Frobenius twist, and let $\pi^{\mathrm{tame}}_1(U_i,\ast)^{(m)}$ be the geometrically maximal $m$-step solvable quotient of the tame fundamental group. The weak bi-anabelian form of the $m$-step solvable Grothendieck conjecture. There exists a pair $(n_1,n_2)$ of non-negative integers with $n_1n_2=0$ such that

$$U_1(n_1)\underset{k}{\simeq}U_2(n_2)\Longleftrightarrow\pi^{\mathrm{tame}}_1(U_1,\ast)^{(m)}\underset{G_k}{\simeq}\pi^{\mathrm{tame}}_1(U_2,\ast)^{(m)}.$$

If, moreover, $U_{1,\overline{k}}$ does not descend to a curve over $\overline{\mathbb F}_p$, then the pair $(n_1,n_2)$ is unique. This formulation accounts for the fact that relative Frobenius twists need not be isomorphic as schemes although they induce isomorphisms on étale fundamental groups. Its status is presented as a conjecture in the source.