Xu–Yip–Zhang conjecture on uniform witness families

Let $[n]=\{1,\ldots,n\}$, let $0\leqslant s\leqslant d$, and let $\mathcal{F}\subseteq\binom{[n]}{d+1}$. An $s$-witness family is a family such that for every $F\in\mathcal{F}$ there exists a set $B_F\subseteq F$ with $|B_F|=s$ and

$$F\cap F'\neq B_F$$

for every $F'\in\mathcal{F}$. Xu–Yip–Zhang conjecture. If $n\geqslant 2(d+1)$ and $\mathcal{F}$ is an $s$-witness family, then

$$|\mathcal{F}|\leqslant\binom{n-1}{d}.$$

The case $s=0$ is the Erdős–Ko–Rado theorem, and the cases $s\in\{1,d\}$ were proved in the cited work. The source proves the conjecture for $s\leqslant d/2$, while constructing non-star families of the conjectured extremal size for $d/2<s\leqslant d-1$; the full statement remains unresolved there.