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

Let $k$ be a finitely generated field of characteristic zero with absolute Galois group $G_k$, let $m$ be a non-negative integer, and let $U_i$ be hyperbolic curves over $k$ of type $(g_i,r_i)$ for $i=1,2$. For each $i$, write $\pi^{\acute{e}\mathrm{t}}_1(U_i,\ast)^{(m)}$ for the geometrically maximal $m$-step solvable quotient of the étale fundamental group, viewed over $G_k$. The weak bi-anabelian form of the $m$-step solvable Grothendieck conjecture.

$$U_1\underset{k}{\simeq}U_2\Longleftrightarrow \pi^{\acute{e}\mathrm{t}}_1(U_1,\ast)^{(m)}\underset{G_k}{\simeq}\pi^{\acute{e}\mathrm{t}}_1(U_2,\ast)^{(m)}.$$

This is the weak form because it asserts reconstruction only at the level of objects and does not impose functoriality. The paper presents results for this conjecture in the genus-zero hyperbolic case.