Bounded-excess connectivity conjecture for Tverberg partition graphs

Let $S$ be a finite set of points in $\mathds{R}^d$, let $r$ be a positive integer, and let $G_T[S,r]$ denote the Tverberg $r$-partition graph of $S$. Let

$$\operatorname{Tv}(d,r)=(d+1)(r-1)+1.$$

The bounded-excess connectivity conjecture. There exists a constant $k(d)$ depending only on $d$ such that $G_T[S,r]$ is connected whenever

$$|S|\geq \operatorname{Tv}(d,r)+k(d),$$

with $k(1)=1$. The authors motivate this by the general connectivity bound and computational observations in the plane, and conjecture that the bound $3\operatorname{Tv}(d,r)-1$ is not optimal; the source does not resolve the existence of such a dimension-dependent constant.