## 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