Reference for the nearby Lagrangian conjecture for $T^*S^1$

I am looking for a reference for the proof of the nearby Lagrangian conjecture of $ T^*S^1$ , that is, that every exact and compact Lagrangian submanifold of the cylinder is Hamiltonianly isotopic to the zero section in $ T^*S^1$ .

The corresponding result for $ T^*S^2$ can be found here https://projecteuclid.org/download/pdf_1/euclid.ajm/1331663450