It is a longstanding open question to characterize those metric spaces that admit a distance-preserving embedding into a $\mathrm{CAT}(\kappa )$ space. On the other hand, among all geodesic metric spaces, $\mathrm{CAT}(\kappa )$ spaces can be characterized by various simple conditions. Gromov's $\mathrm{Cycl}_4 (\kappa )$ condition, which is equivalent to the so-called $\mathrm{CAT}(\kappa )$ $4$-point condition, is one of them. In this talk, we study the difference between the class of metric spaces that admit a distance-preserving embedding into a $\mathrm{CAT}(\kappa )$ space and the class of metric spaces that satisfy the $\mathrm{Cycl}_4 (\kappa )$ condition. We show that an analogue of Reshetnyak's majorization theorem holds for all metric spaces that satisfy the $\mathrm{Cycl}_4 (\kappa )$ condition. We also show that a metric space that contains at most five points admits a distance-preserving embedding into a $\mathrm{CAT}(0)$ space if and only if it satisfies the $\mathrm{Cycl}_4 (0)$ condition.