Higher-dimensional pandichotomy of complete bipartite graphs

Characterize the complete bipartite graphs that are pandichotomous in Euclidean dimension $k$ for every $k\ge 3$.

Background

The paper completely characterizes the complete bipartite graphs that are pandichotomous in the plane: $K_{m,n}$ is pandichotomous in the plane exactly when $m\le 3$ or when $m=4$ and $n\le 6$. It does not provide the corresponding characterization in dimensions $k\ge 3$.

The conclusion explicitly asks for a characterization in higher dimensions, including both the general complete bipartite case and, by implication, the broader bipartite setting.