Giroux–Pardon conjecture on regular Lagrangians in Lefschetz fibrations

Let $W$ be a Weinstein domain and let $L\subset W$ be a regular Lagrangian submanifold with nonempty boundary.

Giroux–Pardon conjecture. There is a Weinstein Lefschetz fibration

$$p:W\to\mathbb{D}^2$$

such that $p(L)$ is an arc with one endpoint on a critical value of $p$ and the other endpoint on $\partial\mathbb{D}^2$.

This is a relative analogue of the Giroux–Pardon existence theorem for Weinstein Lefschetz fibrations, conjectured to follow from the approximately holomorphic techniques used in that theorem. The conjecture is established by the present paper using Morse-theoretic methods and convex hypersurface theory.