Uniqueness of conformal compactification

The conformal compactification $N=bar{iota(X)}$ of a space $X$ is defined to be the closure of the image of a conformal embedding $iota$ which is compact, and every conformal map from an open subset of $X$ to $X$ has a continuation as a conformal map from $N$ to $N$, see page 28 in "A mathematical introduction to conformal field theory" by Prof. Schottenloher. It was further claimed that conformal compactification is unique if it exists in the book. Does anyone know a proof of it?

In particular, I don’t even know how to make one to one correspondence between points in $N_1-iota_1(X)$ and $N_2-iota_2(X)$ with $N_1$ and $N_2$ being two conformal compactifications. A point in $N_1-iota_1(X)$ can be taken as a limit of a sequence in $iota_1(X)$, but this sequence may not be a convergent one in $X$ or $iota_2(X)$ due to conformal factors.