Separating-line conjecture for two point sets in the Cartesian plane

Let $X_{0}$ and $X_{1}$ be two sets of points in the Cartesian plane. Assume that no two line segments determined by four distinct points, with exactly two points from one set, intersect unless the endpoints of one segment belong to different sets. Let $A$ and $B$ be the two disjoint regions into which a line $l_{1}$ divides the plane, excluding points on the line. Separating-line conjecture. There exists at least one line $l_{1}$ such that $X_{0} \subset A$ and $X_{1} \subset B$; no line segment joining two points of the same set meets $l_{1}$; and every line segment joining points from different sets meets $l_{1}$. This conjecture is introduced as a geometric foundation for the proposed binary-classification algorithm; the source provides no resolution or proof status beyond presenting it as a conjecture.