Hilton–Milner codegree-squared-sum conjecture for intersecting families

Let $[n]=\{1,\ldots,n\}$, let $\binom{[n]}{k}$ denote the family of $k$-subsets of $[n]$, and let $\mathcal{F}\subset\binom{[n]}{k}$ be a nontrivial intersecting family, meaning that every two members intersect and the family is not contained in a $1$-star. Let $\mathcal{H}(n,k,1)$ be the Hilton–Milner family, and let $\mathcal{A}(n,3,1)$ be the other extremal family appearing in the $k=3$ case.

Hilton–Milner codegree-squared-sum conjecture. For $k\ge3$ and $n>2k$,

$$\operatorname{co}_2(\mathcal{F})\le\operatorname{co}_2(\mathcal{H}(n,k,1)).$$

For $k\ge4$, equality holds if and only if $\mathcal{F}$ is isomorphic to $\mathcal{H}(n,k,1)$; for $k=3$, equality holds if and only if $\mathcal{F}$ is isomorphic to $\mathcal{H}(n,3,1)$ or to $\mathcal{A}(n,3,1)$. The paper notes that the $k=3$ equality of the two candidate families can be checked directly, while the general Hilton–Milner-type result remains open.