NP-hardness conjecture for the bicolored MST crossing number

Let $P$ be a finite point set in the plane, and let $\operatorname{cr-MST}(P)$ denote its bicolored minimum spanning tree crossing number. The NP-hardness conjecture. Finding $\operatorname{cr-MST}(P)$ is NP-hard.

This concerns the computational complexity of the bicolored MST crossing number. The paper presents NP-hardness as a conjecture and gives no proof or resolution.