The vanishing-metric conjecture for Legendrian lifts of exact Lagrangians

Let $(M,d\lambda)$ be an exact symplectic manifold, and let $L$ be a closed exact Lagrangian submanifold with vanishing Chekanov–Hofer metric. Let $\Lambda$ be the Legendrian lift of $L$ to the contactization $(M\times\mathbb{R},dz+\lambda)$. Vanishing-metric conjecture. The Legendrian lift $\Lambda$ has vanishing Shelukhin–Chekanov–Hofer metric. This is presented as an expected generalization of the preceding theorem, which proves the claim under the assumptions of a proposition concerning a pair of exact, displaceable Lagrangians. No resolution of this generalization is given in the source.