Frankl–Kiselev–Kupavskii conjecture on symmetric differences of intersecting families

Let $[n]=\{1,2,\dots,n\}$, and let $\binom{[n]}{k}$ denote the family of all $k$-element subsets of $[n]$. A family $\mathcal{F}\subseteq\binom{[n]}{k}$ is intersecting if $F\cap F'\neq\emptyset$ for all $F,F'\in\mathcal{F}$. For sets $F,G$, write $F\triangle G=(F\cup G)\setminus(F\cap G)$, and define

$$\mathcal{SD}(\mathcal{F})=\{F\triangle G:F,G\in\mathcal{F}\}.$$

Frankl–Kiselev–Kupavskii conjecture. If $\mathcal{F}\subseteq\binom{[n]}{k}$ is intersecting and $n>10k$, then

$$|\mathcal{SD}(\mathcal{F})|\le\sum_{\ell=0}^{k-1}\binom{n-1}{2\ell}.$$

The right-hand side is attained by a full star, so the conjecture asserts that full stars maximize the number of symmetric differences among intersecting $k$-uniform families in the stated range. The supplied text gives the conjecture's origin but no resolution, so its status is open.