Grothendieck's Hom conjecture for hyperbolic curves

Let $k$ be finitely generated over $\mathbb{Q}$. Let $T/k$ be a smooth variety, and let $X/k$ be a smooth, proper, geometrically connected hyperbolic curve. Grothendieck's Hom conjecture. The natural map

$$\operatorname{Hom}_{k}(T,X)\to\operatorname{Hom-ext}_{G_k}(\pi_1(T,t),\pi_1(X,x))$$

is a bijection. This is one of Grothendieck's two forms of the main conjecture of anabelian geometry; the source notes that its weaker dominant-morphism form was proved by Mochizuki, while the full statement is not resolved here.