Three-space compatible optimal correspondences conjecture

Let $\mathcal{M}$ denote the collection of compact metric spaces, and for $i,j\in\{1,2,3\}$ let $e_{ij}:X_1\times X_2\times X_3\to X_i\times X_j$ be the coordinate projection. A correspondence between two metric spaces is a relation whose projections onto both factors are surjective; it is optimal when it attains the Gromov-Hausdorff distance. Compatible optimal-correspondences conjecture. Given three compact metric spaces $X_1,X_2,X_3\in\mathcal{M}$, there exists $R\subseteq X_1\times X_2\times X_3$ such that $e_{ij}(R)$ is an optimal correspondence between $X_i$ and $X_j$ for all $i,j\in\{1,2,3\}$. This asks whether pairwise optimal correspondences can always be realized simultaneously as projections of one three-way relation; it is posed among the paper's related unsolved problems.