In the paper Curves and their Fundamental Groups written by Gerd Faltings, Mochizuki's proof of Grothendick's conjecture on anabelian curves is explained.

In the proof, he shows that for two hyperbolic curves $X$ and $Y$, if there exists an isomorphism between the algebraic fundamental groups $\pi_1(X)$ and $\pi_1(Y)$ then these 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 $\pi_1(X)$?