Tightness of the quadratic Gromov–Wasserstein correspondence-plan theorem

Let $\mu$ and $\nu$ be probability measures in the setting of the quadratic Gromov–Wasserstein problem. An optimal correspondence plan is a transport plan attaining the optimum for the quadratic Gromov–Wasserstein objective, and a plan is a map when it is induced by a measurable transport map; an anti-map is the analogous graph of a map with reversed orientation.

Tightness conjecture. Theorem is tight: there exist $\mu$ and $\nu$ for which optimal correspondence plans for the quadratic Gromov–Wasserstein problem are not maps, but are instead a union of two graphs, namely the graphs of two maps or of one map and one anti-map. This can occur even when $\mu$ has a density.

The claim gives empirical evidence that the theorem cannot be strengthened to guarantee that an optimal correspondence plan is a map, despite the classical optimal-transport condition that absolute continuity of $\mu$ yields an optimal transport map.