Full-range low-degree vertex conjecture for intersecting hypergraphs

Let $[n]=\{1,2,\ldots,n\}$, and let $\mathcal{F}\subset \binom{[n]}{k}$ be an intersecting family, meaning that any two members of $\mathcal{F}$ intersect. The degree of a vertex is the number of members of $\mathcal{F}$ containing it.

Full-range low-degree vertex conjecture. If $n>2k$, then at least $n-2k$ vertices have degree at most

$$\binom{n-2}{k-2}.$$

The paper proves this conclusion under the stronger hypothesis $n\geq 6k$, while the Huang–Zhao theorem gives one such vertex for $n>2k$. The conjecture asks whether the bound of $n-2k$ vertices holds throughout the full range $n>2k$.