Regular Lagrangian Conjecture in Weinstein manifolds

Prove the regular Lagrangian conjecture asserting that for any Weinstein manifold (W, dθ), every exact closed Lagrangian submanifold L ⊂ W can be realized as a subset of the skeleton of W after an appropriate choice of Weinstein structure.

Background

The paper discusses connections between exact Lagrangian submanifolds in a Weinstein domain and representations of the Chekanov–Eliashberg algebra. In this context, the authors recall a central conjecture in the field—often referred to as the regular Lagrangian conjecture—asserting that any exact closed Lagrangian can be made to lie inside the Weinstein skeleton under a suitable choice of Weinstein structure.

They note that while their methods can homotope Lagrangians to configurations closely related to the skeleton (allowing regular exact Lagrangian homotopies that introduce double points), they currently lack the technology to prove the full conjecture. This underscores the conjecture’s status as an important open problem motivating parts of their approach.