Grothendieck's anabelian reconstruction conjecture

Let $k$ be a field finitely generated over the prime field, let $G_k$ be its absolute Galois group, and let $U$ be an anabelian variety over $k$ with a geometric point $\ast$. Its étale fundamental group is $\pi^{\acute{e}\mathrm{t}}_1(U,\ast)$, equipped with its natural projection

$$\pi^{\acute{e}\mathrm{t}}_1(U,\ast)\to G_k.$$

Grothendieck's conjecture. The variety $U$ may be reconstructed group-theoretically from $\pi^{\acute{e}\mathrm{t}}_1(U,\ast)$ and this projection. The conjecture for hyperbolic curves is solved, but the general formulation here concerns anabelian varieties whose definition is not yet completely established.