The strong Arnol'd chord conjecture for the boundary of a uniformly convex domain in ${\mathbb{R}}^4$

Following the idea of Jungsoo Kang and Jun Zhang, we prove the strong Arnol’d chord conjecture for the boundary of a uniformly convex domain in ${\mathbb{R}}^4$, using an ellipsoid embedding construction due to Oliver Edtmair. We prove a general structural result for Legendrians $L$ which are eventually equivariantly essential (E3), in the sense that the $k$th Gutt-Hutchings capacity $c_k(D^{\ast }TL)$ is infinite for $k$ large enough. We show that any E3 Legendrian in the boundary of a Liouville domain $\mathrm{\Omega }$ bounds a chord of length at most $\mathrm{lim\hspace{0.17em}inf}{c}_k(\mathrm{\Omega })\mathrm{/}k$.