Winding partitions conjecture for drawings of complete graphs

Let $K_{3(q-1)+1}$ be the complete graph on $3(q-1)+1$ vertices, and consider any drawing of this graph in the plane. For a triangle of edges and a point not on it, its winding number about the point is the usual winding number of the closed triangular curve. Winding partitions conjecture. Every drawing has either $q-1$ pairwise disjoint triangles of edges and a vertex $v$, or $q-2$ pairwise disjoint triangles of edges and an intersection point $p$ of two edges, such that the winding number about $v$, respectively about $p$, of every triangle is nonzero. The conjecture is connected to the $d=2$ case of the general Tverberg-partition problem through geometric graph theory. The paper presents it as an equivalent formulation of that case; no resolution is given here.