Bukh–Matoušek–Nivasch conjecture on simplicial depth

For a finite set $P$ of $n$ points in $\mathbb{R}^d$, define its simplicial depth as the maximum number of $d$-simplices spanned by $P$ that contain a common point. Bukh–Matoušek–Nivasch conjecture. The simplicial depth of $P$ is at least $\left(\left\lceil\frac{n}{d+1}\right\rceil\right)^{d+1}$. The source describes this as a long-standing problem and gives partial bounds in several dimensions.