Frankl–Wang's $3/7$ conjecture on maximum-degree ratios

Let $[n]=\{1,2,\dots,n\}$, and let $\binom{[n]}{k}$ denote the family of all $k$-subsets of $[n]$. A family $\mathcal{F}\subseteq\binom{[n]}{k}$ is intersecting if $F\cap F'\neq\emptyset$ for all $F,F'\in\mathcal{F}$. Define

$$\Delta(\mathcal{F})=\max_{x\in[n]}|\{F\in\mathcal{F}:x\in F\}|,\qquad \varrho(\mathcal{F})=\frac{\Delta(\mathcal{F})}{|\mathcal{F}|}.$$

Frankl–Wang's $3/7$ conjecture. If $\mathcal{F}\subseteq\binom{[n]}{k}$ is intersecting, $n>100k$, and $|\mathcal{F}|>\binom{n-3}{k-3}$, then

$$\varrho(\mathcal{F})\ge \frac{3}{7}.$$

The ratio $\varrho(\mathcal{F})$ measures concentration on the most popular element, so the conjecture gives a quantitative lower bound on this concentration for sufficiently large intersecting families. The source presents it as a conjecture of Frankl and Wang; no resolution is supplied.