I had a geometry class which was proctored using the Moore method, where the questions were given but not the answers, and the students were the source of all answers in the class. One of the early questions which we never solved is listed in the title.
In this case, use any reasonable definition of "between-ness". I believe the definition we used was "$B$ is between $A$ and $C$ if and only if $|AB|+|BC|=|AC|$". A collineation is a mapping where every line is mapped to a line. A mapping is a function that operates over the set of points within the given space and returns points in the given space.
When we were studying this question, we managed to get to the point that a between-ness mapping must map lines to line segments. In particular, we had managed to show that for every $A-B-C$ (read: "$B$ is between $A$ and $C$") in the mapped space by applying between-ness preserving mapping $m$, we could guarantee that pre-images $P_A, P_C$ such that $m(P_A)=A,m(P_C)=C$ implies the existence of pre-image $P_B$ such that $P_A-P_B-P_C$ and $m(P_B)=B$. I have never seen any full proof of the title statement.
I would like to read any hints that are known for solving this question. Feel free to completely solve it, but please hide the full solution in such a way that I can start with your hint(s) and have an opportunity to finish the solution for myself.