Linear-range low-degree conjecture for t-intersecting hypergraphs

Let $[n]=\{1,2,\ldots,n\}$, and let $\mathcal{F}\subset \binom{[n]}{k}$ be a $t$-intersecting family, meaning that any two members of $\mathcal{F}$ have intersection of size at least $t$. Let $d_{k+2}(\mathcal{F})$ denote the $(k+2)$nd largest vertex degree in $\mathcal{F}$.

Linear-range low-degree conjecture. There is an absolute constant $c$ such that, whenever $n\geq ckt$,

$$d_{k+2}(\mathcal{F})\leq \binom{n-t-1}{k-t-1}.$$

The paper proves a related bound for $n\geq \binom{t+2}{2}k^2$, with at least $n-k-1$ vertices meeting the corresponding degree threshold. This stronger conjecture predicts the same type of control in the linear range $n\geq ckt$.