Kalai's cascade consequence for higher Tverberg partitions

Let $d,r$ be positive integers and let $k$ be a non-negative integer. A Tverberg $(r+1)$-partition of a finite point set is a partition into $r+1$ parts whose convex hulls have a common point. Kalai's cascade consequence. If $S$ is a set of $(r-1)(d+1)+k+2$ points in $\mathbb{R}^d$ and

$$\dim(T_r(S))\leq k,$$

then $S$ has at least one Tverberg $(r+1)$-partition. This is presented as a direct consequence of Kalai's cascade conjecture and remains unconfirmed except when $r=1$ or $d\leq 2$; the case $r=1$ is Radon's theorem.