The minimum-degree conjecture for intersecting uniform families

Let $[n]=\{1,\ldots,n\}$, let $k>d\geq 0$, and let $\mathcal{F}\subset\binom{[n]}{k}$ be an intersecting family, meaning that any two members of $\mathcal{F}$ have nonempty intersection. Write $\delta_d(\mathcal{F})$ for the minimum $d$-degree of $\mathcal{F}$.

Minimum-degree conjecture. For every $k>d\geq 0$ and $n\geq 2k+1$, an intersecting family $\mathcal{F}\subset\binom{[n]}{k}$ always satisfies

$$\delta_d(\mathcal{F})\leq\binom{n-d-1}{k-d-1}.$$

The result proved in the paper establishes the corresponding bound for $n\geq 2k+2d-3$. The conjecture is stated to have been verified for $d\in\{0,1,2\}$, while the full range $n\geq 2k+1$ remains open.