  Simple drawings are drawings of graphs in the plane or on the sphere such
that vertices are distinct points, edges are Jordan arcs connecting their
endpoints, and edges intersect at most once (either in a proper crossing or in
a shared endpoint). Simple drawings are generalized twisted if there is a point
$O$ such that every ray emanating from $O$ crosses every edge of the drawing at
most once and there is a ray emanating from $O$ which crosses every edge
exactly once. We show that all generalized twisted drawings of $K_n$ contain
exactly $2n-4$ empty triangles, by this making a substantial step towards
proving the conjecture that this is the case for every simple drawing of $K_n$.
