  We study symplectic rigidity phenomena for fibers in cotangent bundles of
Riemann surfaces. Our main result can be seen as a generalization to open
Riemann surfaces of arbitrary genus of work of Eliashberg and Polterovich on
the Nearby Lagrangian Conjecture for $T^* \mathbb{R}^2$. As a corollary, we
answer a strong version in dimension $2n=4$ of a question of Eliashberg about
linking of Lagrangian disks in $T^* \mathbb{R}^n$, which was previously
answered by Ekholm and Smith in dimensions $2n \geq 8$.
