Extend the low-degree vertex-count bound to the full range n>2k

Prove that every intersecting k-uniform hypergraph F\subset\binom{[n]}{k} with n>2k has at least n-2k vertices whose degrees are at most \binom{n-2}{k-2}.

Background

The paper proves that, for n\geq 6k, every intersecting k-graph on n vertices has at least n-2k vertices of degree at most \binom{n-2}{k-2}. This strengthens the Huang--Zhao theorem, which guarantees only one vertex with degree at most that quantity for all n>2k.

The conjecture asks whether the stronger count of n-2k low-degree vertices remains valid throughout the entire nontrivial intersecting range n>2k, rather than only in the range established by the paper.