Loose Legendrian rigidity under contact homeomorphisms

Let $M$ be a smooth manifold and let $\Lambda\subseteq S^*M$ be a proper loose Legendrian embedding. Consider contactomorphisms $\varphi_n\in\operatorname{Cont}_0(S^*M,\xi_{\mathrm{std}})$ such that $\varphi_n\to\varphi_\infty$ in the $C^0$-topology, where $\varphi_\infty$ is a homeomorphism. Assume that $\varphi_\infty(\Lambda)$ is also smooth.

Loose Legendrian rigidity conjecture. The Legendrian $\varphi_\infty(\Lambda)$ is still loose.

This would rule out a known non-loose Legendrian in a cosphere bundle arising as the $C^0$-limit of a loose Legendrian. The conjecture is motivated by invariance properties of sheaf categories and Legendrian contact homology; the source indicates that the corresponding higher-dimensional rigidity statement is not known.