Existence of exact Lagrangian embeddings underlying cotangent contact inclusions

Let $L$ and $N$ be simply connected manifolds, and let $ST^*L$ denote the unit cotangent sphere bundle of $L$, viewed as a contact manifold embedded in $T^*N$. An exact Lagrangian embedding $L\hookrightarrow T^*N$ is a Lagrangian embedding for which the ambient Liouville one-form restricts to an exact one-form on $L$; a Weinstein neighbourhood is a standard symplectic neighbourhood of this embedding. The cotangent contact inclusion conjecture. Every exact contact inclusion

$$ST^*L\subset T^*N$$

arises as the boundary of a Weinstein neighbourhood of an exact Lagrangian embedding $L\hookrightarrow T^*N$. The supplied text gives no evidence establishing or refuting this assertion, so its resolution status is left open.