The Andoni–Naor–Neiman inequalities and isometric embeddability into a CAT(0) space

Andoni–Naor–Neiman [Snowflake universality of Wasserstein spaces, Ann. Sci. Éc. Norm. Supér. 51 (2018) 657–700] established a family of quadratic metric inequalities that hold true in every CAT(0) space. As stated in their paper, this family seems to include all previously used quadratic metric inequalities that hold true in every CAT(0) space. We prove that there exists a metric space that satisfies all inequalities in this family but does not admit an isometric embedding into any CAT(0) space. More precisely, we prove that the 6-point metric space constructed by Nina Lebedeva, which does not admit an isometric embedding into any CAT(0) space, satisfies all inequalities in this family.