  Andoni, Naor and Neiman (2018) established a family of quadratic metric
inequalities that hold true in every $\mathrm{CAT}(0)$ space. As stated in
their paper, this family seems to include all previously used quadratic metric
inequalities that hold true in every $\mathrm{CAT}(0)$ 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 $\mathrm{CAT}(0)$ space. More
precisely, we prove that the $6$-point metric space constructed by Nina
Lebedeva, which does not admit an isometric embedding into any
$\mathrm{CAT}(0)$ space, satisfy all inequalities in this family.
