Prove the linear-threshold bound for t-intersecting hypergraphs

Establish that there exists an absolute constant c such that, for every t-intersecting k-uniform hypergraph F\subset\binom{[n]}{k} and every n\geq ckt, the (k+2)th largest degree satisfies d_{k+2}(F)\leq\binom{n-t-1}{k-t-1}.

Background

Theorem 5 proves that every t-intersecting k-graph has d_{k+2}(F)\leq\binom{n-t-1}{k-t-1} under the substantially stronger condition n\geq\binom{t+2}{2}k^2, with k>t\geq 1.

The conjecture seeks to reduce this quadratic-type threshold in k and t to a linear threshold n\geq ckt, with c an absolute constant, while retaining the same degree bound.