For any metric space $X$ and $\varepsilon>0$, let $$\text{cov}(X,\varepsilon)=\min\{n\,|\,X\text{ has a cover by }n\text{ many closed }\varepsilon\text{-balls}\},$$

be the ordinary covering numbers. For any $D>0$ and $N:(0,\infty)\rightarrow \mathbb{N}$, let $$U(D,N)=\{X\,|\,X\text{ is a compact metric space, diam}(X)\leq D,\,\text{cov}(X,\varepsilon)\leq N(\varepsilon)\},$$

be the set of compact metric spaces with uniformly bounded diameter and covering numbers. Gromov's compactness theorem implies that $U(D,N)$ is totally bounded with respect to the Gromov-Hausdorff distance for any $D$ and $N$, so in particular $\text{cov}_{GH}(U(D,N),\varepsilon)$ is always finite.

Are there any known bounds on $\text{cov}_{GH}(U(D,N),\varepsilon)$ in terms of $N$ (everything clearly scales with $D$, so setting $D=1$ is sufficient)?