The strong Arnol'd chord conjecture for uniformly convex domains

Let $\Omega$ be a uniformly convex domain in $\mathbb{R}^{2n}$ with $n>1$, and let $c_k(\Omega)$ denote its Gutt--Hutchings capacities. The first capacity $c_1(\Omega)$ is the systolic action associated with the domain. Strong Arnol'd chord conjecture.

$$\liminf_{k\to\infty}\frac{c_k(\Omega)}{k}<c_1(\Omega).$$

This inequality is noted to hold when $n=2$, by the cited lemma; the conjecture concerns uniformly convex domains in all dimensions $2n$ with $n>1$.