The strong bi-anabelian m-step solvable Grothendieck conjecture

Let $k$ be a field with absolute Galois group $G_k$, let $U_1,U_2$ be the curves under consideration, and let $m,n$ be non-negative integers. For each $r$, write $\pi^{\acute{e}\mathrm{t}}_1(U_i,\ast)^{(r)}$ for the corresponding geometrically maximal $r$-step solvable quotient. The groups of $k$-isomorphisms and $G_k$-equivariant isomorphisms are denoted by $\mathrm{Isom}_k$ and $\mathrm{Isom}_{G_k}$, respectively, and $\mathrm{Inn}$ denotes inner automorphisms. The strong bi-anabelian form of the $m$-step solvable Grothendieck conjecture. Fix a non-negative integer $n$. The image of

$$\mathrm{Isom}_{k}(U_{1},U_{2})\to \mathrm{Isom}_{G_{k}}(\pi^{\acute{e}\mathrm{t}}_{1}(U_{1},\ast)^{(m)},\pi^{\acute{e}\mathrm{t}}_{1}(U_{2},\ast)^{(m)})/\mathrm{Inn}(\pi^{\acute{e}\mathrm{t}}_{1}(U_{2,k^{\mathrm{sep}}},\ast)^{m})$$

coincides with the image of

$$\mathrm{Isom}_{G_{k}}(\pi^{\acute{e}\mathrm{t}}_{1}(U_{1},\ast)^{(m+n)},\pi^{\acute{e}\mathrm{t}}_{1}(U_{2},\ast)^{(m+n)})/\mathrm{Inn}(\pi^{\acute{e}\mathrm{t}}_{1}(U_{2,k^{\mathrm{sep}}},\ast)^{m+n}).$$

The strong form refines the weak form by incorporating functoriality. The source discusses previous results and the role of coverings, but does not resolve the conjecture in this general formulation.