Reay's relaxed Tverberg conjecture

Let $T(d,r,k)$ be the least positive integer $n$ such that every set of $n$ points in $\mathbb{R}^d$ admits a partition into $r$ disjoint parts whose convex hulls have nonempty intersection in every $k$-member subfamily. Reay's relaxed Tverberg conjecture. For all $2\leq k\leq r$, $T(d,r,k)=(r-1)(d+1)+1$. The source reports that this is known in several parameter ranges and open in general.