Strong nearby Lagrangian conjecture (contractibility of L(M))

Prove the strong nearby Lagrangian conjecture for any closed connected manifold M: the space L(M) of closed connected exact Lagrangian submanifolds in the cotangent bundle T^*M is contractible.

Background

Beyond path-connectedness, a stronger expectation is that L(M) is contractible (the strong nearby Lagrangian conjecture). This would imply, in particular, that all higher homotopy groups of L(M) vanish. The authors note that no counterexamples are known and only low-dimensional cases are currently established.

The techniques developed in the paper yield strong constraints on the parametrised Whitehead torsion and in certain cases show weak nullhomotopy of associated maps, but do not prove the full contractibility statement.