Consider a sample $X_1,...,X_n$ of i.i.d. random variables from distribution $P\in\mathcal{P}$, where $\mathcal{P}$ is the set of all distributions over $X_1 \in \mathbb{R}^d$ for some positive integer $d$. Let $\mathcal{F}$ be an arbitrary class of binary functions, i.e. $f:\mathbb{R}^d \rightarrow \{0,1\}$, and let $\mathcal{F}(X_1,\dots,X_n)$ be the restriction of $\mathcal{F}$ to the sample $X_1,\dots,X_n$, that are all the unique functions from $X_1,...,X_n$ to $\{0,1\}^n$, i.e. \begin{align} \mathcal{F}(X_1,\dots,X_n) = \{(f(X_1),\dots,f(X_n)):f\in\mathcal{F}\}. \tag{1} \end{align} The restriction $(1)$ is basically a set of binary vectors of length $n$. On the basis of $(1)$ one can define the following capacity concepts, \begin{align} {H}^{\mathcal{F}}(n) &= \mathbb{E}\{\ln\lvert \mathcal{F}(X_1,\dots,X_n) \rvert\}, \\[1em] \widetilde{H}^{\mathcal{F}}(n) &= \ln\mathbb{E}\{\lvert \mathcal{F}(X_1,\dots,X_n) \rvert\}, \\[1em] G^{\mathcal{F}}(n) &= \sup_{x_1,\dots,x_n}\ln \lvert\mathcal{F}(x_1,\dots,x_n)\rvert \end{align} (provided that the restriction $(1)$ is measurable, which we will assume). These are called the VC entropy, annealed entropy and growth function, respectively. Here $\lvert \cdot\rvert$ denotes the cardinality of the considered set.
According to Vapnik (1995), The Nature of Statistical Learning Theory, these capacity concepts are such that, for all $n$,
\begin{align}
{H}^{\mathcal{F}}(n) \leq \widetilde{H}^{\mathcal{F}}(n) \stackrel{(a)}{\leq} {G}^{\mathcal{F}}(n),
\end{align}
which is clear. However, I am wondering whether there exists a combination of a function class $\mathcal{F}$ and a distribution $P\in\mathcal{P}$, such that,
\begin{align}
\widetilde{H}^{\mathcal{F}}(n) = G^{\mathcal{F}}(n), \tag{2}
\end{align}
for some $n > 1$, that is, should inequality $(a)$ not be strict? It seems to me that this equality cannot hold true, since the i.i.d. assumption makes it impossible to obtain the 'worst-case sequence' with a high probability (ideally probability 1). However, to my understanding, the existence of a class $\mathcal{F}$ and a distribution $P$ such that the equality holds, seems to be critically important for the claim made by Vapnik (1995), labelled the third milestone in learning theory, that,
\begin{align}
\lim_{n\rightarrow\infty}\frac{{G}^{\mathcal{F}}(n)}{n} =0,
\end{align}
is a necessary and sufficient condition for the consistency of the ERM classification rule. I understand that this is a very niche part of mathematics, however I do hope that somebody can shed some light on the situation presented above. Does there exist a class $\mathcal{F}$ and a distribution $P$ such that equality $(2)$ holds true?