The bipartite rigidity conjecture for complete graphs

Consider the complete bipartite graph $K_{m,n}$ and a realization $(A,B)\in R$ of $K_{m,n}$ in $\mathbb{R}^d$, where $A$ and $B$ are the two point classes. Let $K(A,B)$ denote the corresponding framework.

Bipartite rigidity conjecture. The framework $K(A,B)$ is rigid in $\mathbb{R}^d$, unless either the points in $A$ or the points in $B$ are contained in a hyperplane in $\mathbb{R}^d$.

This is presented as a step toward the distinct-distances conjecture and as an extension of a result of Bolker and Roth. The source does not provide a resolution, so the conjecture remains open.