3 problems

$C^0$-flux conjecture. The Hamiltonian diffeomorphism group $\operatorname{Ham}(M,\omega)$ is $C^0$-closed in $\operatorname{Symp}_0(M,\omega)$.

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

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