The discrete filling-area density conjecture

Let $C_n$ be the cycle graph on $n$ vertices, and let $D(n;\epsilon)$ denote the minimum number of vertices in a $(1-\epsilon)$-Lipschitz filling of $C_n$. Define

$$D^*\stackrel{\text{def}}{=}\liminf_{\epsilon\to 0^+}\liminf_{n\to\infty}\frac{D(n;\epsilon)}{n^2}.$$

The discrete filling-area density conjecture.

$$D^*<\frac{1}{\pi\sqrt{3}}.$$

The paper's bounds show that $D^*\leq 1/(\pi\sqrt{3})$ and that the strict inequality would separate the discrete problem from Gromov's continuous filling-area conjecture. The authors believe the strict inequality strongly, but do not provide a construction proving it.