Hausdorff-boundedness and Wasserstein-realizability of Gromov-Wasserstein geodesics

Let $p\in[1,\infty)$, and let an $\ell^p$-Gromov-Wasserstein geodesic be a geodesic for the $\ell^p$-Gromov-Wasserstein distance. A geodesic is Hausdorff-bounded if it satisfies the corresponding uniform Hausdorff control, and it is Wasserstein-realizable if it arises from a Wasserstein geodesic under the Gromov-Wasserstein construction. Gromov-Wasserstein geodesic conjecture. For every $p\in(1,\infty)$, any $\ell^p$-Gromov-Wasserstein geodesic is Hausdorff-bounded. Also, for every $p\in[1,\infty)$, any $\ell^p$-Gromov-Wasserstein geodesic is Wasserstein-realizable. The paper presents these assertions as an open problem after exhibiting a non-Hausdorff-bounded example for $p=1$ and noting that no such example is known for $p>1$.