The weakened Nearby Lagrangian conjecture

Let $L$ be a closed manifold and let $T^*L$ denote its cotangent bundle. Let $K\subset T^*L$ be an exact Lagrangian submanifold. Assume that some Weinstein neighbourhood of $K$ contains the zero section $L$. Weakened Nearby Lagrangian conjecture. $K$ is Hamiltonian isotopic to $L$. This is proposed as a weakened form of the Nearby Lagrangian conjecture. The paper presents it as a question motivated by rigidity results for Lagrangian submanifolds that contain the zero section in a Weinstein neighbourhood; its resolution is not established here.