In the paper Curves and their Fundamental Groups written by Gerd Faltings, Mochizuki's proof of the Grothendick's Conjecture in Anabelian Curves is explained.

In the proof he shows that for two hyperbolic curves $X$ and $Y$, if there exist an isomorphism between the algebraic fundamental groups $\pi_1(X)$ and $\pi_1(Y)$ then this curves are isomorphic.

My question is: With only the fundamental group $\pi_1(X)$, can the hyperbolic curve $X$ be reconstructed? Specifically can the differential sheaf $\omega_X$ be reconstructed using only the algebraic fundamental group $\pi_1(X)$?