Legendrian Gauss-map rigidity under contact homeomorphisms

Let $(Y,\xi)$ be a contact manifold and let $\Lambda\subseteq(Y,\xi)$ be a Legendrian embedding. Consider contactomorphisms $\varphi_n\in\operatorname{Cont}_0(Y,\xi)$ such that $\varphi_n\to\varphi_\infty$ in the $C^0$-topology, where $\varphi_\infty$ is a homeomorphism. When $\varphi_\infty(\Lambda)$ is smooth, its Lagrangian Gauss map—the map from the Legendrian to $U/O$ determined by its tangent Lagrangian planes—should remain the same:

Legendrian Gauss-map rigidity conjecture. The diagram relating the Lagrangian Gauss maps of $\Lambda$ and $\varphi_\infty(\Lambda)$ commutes:

\begin{tikzcd}[row sep=4pt] \Lambda \ar[dr] \ar[dd, "\varphi_\infty" \left] & \\ & U/O \\ \varphi_\infty(\Lambda) \ar[ur] & \end{tikzcd}

This would extend the known rigidity of Maslov data in cosphere bundles to general contact manifolds. The statement is motivated by the difficulty of cutting off contactomorphisms near closed Legendrians while preserving their $C^0$-distance; its resolution in higher dimensions is not supplied here.