Half-space separation in monophonic convexity

We study half-space separation in the convexity of chordless paths of a graph, i.e., monophonic convexity. In this problem, one is given a graph and two (disjoint) subsets of vertices and asks whether these two sets can be separated by complementary convex sets, called half-spaces. While it is known this problem is $\mathbf{N}\mathbf{P}$-complete for geodesic convexity – the convexity of shortest paths – we show that it can be solved in polynomial time for monophonic convexity.