The homotopy-theoretic Nearby Lagrangian conjecture

Let $N$ and $L$ be closed, simply connected manifolds of the same dimension. The cotangent bundle $T^*N$ carries its natural symplectic structure, and consider the space of Lagrangian embeddings of $L$ into $T^*N$. Homotopy-theoretic Nearby Lagrangian conjecture. The space of Lagrangian embeddings of $L$ in $T^*N$ is contractible if $L$ is diffeomorphic to $N$, and is empty otherwise. This is a weaker homotopy-theoretic version of Arnold's Nearby Lagrangian conjecture, which concerns Lagrangian submanifolds of cotangent bundles and remains open. The conjecture is motivated by recent homotopy-theoretic and Floer-theoretic results.