This is a follow-up to this post.

Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by $$ \kappa_X(\epsilon)\triangleq\sup\left\{ k : \exists x_0,\dots,x_k \in X : \exists r>0, \sqcup_{i=1}^k B(x_i;\epsilon r)\subseteq B(x_0;r) \right\} . $$

If $(\mathbb{R}^D,d_{Euc})$ is a Euclidean space, with $D$ large enough, such that there exists a $C>0$ satisfying $$ K_X(\epsilon) \leq CK_{\mathbb{R}^D}(\epsilon);\qquad \left(\forall \epsilon \in \Big[0,\frac1{3}\Big)\right). $$

Does there necessarily exist a continuous injection of $(X,d_X)$ into a metric ball in $(Y,d)$ of some (finite) radius?

This is a follow-up to this post.

Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by $$ \kappa_X(\epsilon)\triangleq\sup\left\{ k : \exists x_0,\dots,x_k \in X : \exists r>0, \sqcup_{i=1}^k B(x_i;\epsilon r)\subseteq B(x_0;r) \right\} . $$

If $(\mathbb{R}^D,d_{Euc})$ is a Euclidean space, with $D$ large enough, such that there exists a $C>0$ satisfying $$ K_X(\epsilon) \leq CK_{\mathbb{R}^D}(\epsilon);\qquad \left(\forall \epsilon \in \Big[0,\frac1{3}\Big)\right). $$

Does there necessarily exist a continuous injection of $(X,d_X)$ into a metric ball in $(\mathbb{R}^D,d_{Euc})$ of some (finite) radius $M>0$?