Background
Let us consider a Structural Equation Model (SEM): $$X_j:=f_j(\mathbf{PA}_j,N_j), ~ j=1,\ldots,p,$$ where $\mathbf{PA}_j$ are the parents of $X_j$ and $N_1,\ldots,N_d$ are jointly independent noise variables.
The graph $G$ of the SEM is obtained by creating a vertex for each $X_j$ and drawing edges from $\mathbf{PA}_j$ to $X_j, j=1,\ldots,p$.
I read that under the assumption of faithfulness, $G$ can be identified upto its Markov Equivalence Class from the distribution of the SEM. [Lemma 7.2 in Elements of Causal Inference by Jonas Peter et al.]
Question
Is the converse true? Specifically, assuming that $G$ is identifiable from the distribution of the SEM, then is it faithful to $G$?