Kiselev–Kupavskii–Patkós minimum-degree conjecture for union antichains

Let $\mathscr{P}([n])$ be the power set of $[n]$. A family $\mathcal{F}\subseteq\mathscr{P}([n])$ is a $(2\ell+1)$-union antichain if it is an antichain and $\lvert F\cup F'\rvert\leq 2\ell+1$ for every $F,F'\in\mathcal{F}$. Write $\delta(\mathcal{F})$ for its minimum degree.

Kiselev–Kupavskii–Patkós' conjecture. If $1\leq 2\ell+1<n$ and $\mathcal{F}\subseteq\mathscr{P}([n])$ is a $(2\ell+1)$-union antichain, then

$$\delta(\mathcal{F})\leq\binom{n-1}{\ell-1}.$$

The conjecture concerns the minimum degree of union antichains and is equivalent to their conjecture on the diversity of an intersecting antichain. Its resolution is not given in the supplied text.