Chao–Xu–Yip–Zhang's uniform certificate conjecture

Let $n\ge 2(d+1)$ and let $0\le s\le d$. Assume $\mathcal{F}\subseteq {[n]\choose d+1}$, where an $\mathcal{F}$-certificate of a member $F$ is the certificate notion defined in the source, and suppose that every $F\in\mathcal{F}$ has an $\mathcal{F}$-certificate of order $s$.

Chao–Xu–Yip–Zhang's conjecture.

$$|\mathcal{F}|\le {n-1\choose d}.$$

This uniform version of the Frankl–Pach conjecture was proposed as an analogue of the Erdős–Ko–Rado theorem. It is verified for $s\in\{0,d\}$ and for $s=1$ when $n$ is sufficiently large compared with $d$, but remains open in general.