Linear-degree conjecture for Tverberg graphs

Let $q\geq 2$ be a prime power, $d\geq 1$, and let $G$ be a graph on $(d+1)(q-1)+1$ vertices. A graph is a $(d,q)$--Tverberg graph if it has the corresponding Tverberg property for parameters $d$ and $q$. Linear-degree conjecture. There is a constant $k$ such that, whenever the maximal degree of $G$ is less than $kq$, $G$ is a $(d,q)$--Tverberg graph. The conjecture proposes a substantial improvement over the sufficient bound $D(D+1)<q$; its status is unclear from the supplied text.