Optimal minimum-degree conjecture for Tverberg partition graphs

Let $S$ be a set of $n$ points in $\mathds{R}^d$ in strong general position, and let $G_T[S,r]$ be its Tverberg $r$-partition graph. Write $\delta(G_T[S,r])$ for the minimum degree of this graph, and let

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

For $n>\operatorname{Tv}(d,r)$, the minimum-degree conjecture. The minimum degree is

$$\delta(G_T[S,r])=(n+1-\operatorname{Tv}(d,r))(r-1).$$

The paper has established lower and upper degree bounds, and this conjecture asserts that the lower bound is attained for every strongly general-position configuration. Its resolution is not given in the source.