Closure of co-bipartite unit disk graphs under bipartite complementation

Let $G$ be a co-bipartite unit disk graph, meaning that its vertex set can be partitioned into two cliques, and let the bipartite complement of $G$ be the graph obtained by complementing the edges between the two parts while leaving the two cliques unchanged. Closure conjecture. For every co-bipartite unit disk graph, its bipartite complement is also a co-bipartite unit disk graph. The conjecture is motivated by the fact that the bipartite complement preserves the unit disk graph property for the subclass whose cross-part edges are $C_4$-free, while the general closure question remains open.