Existence of codes with closed embedding dimension exceeding base set size

Let $[n]=\{1,\ldots,n\}$, and let $\operatorname{cdim}(\mathcal C)$ denote the closed embedding dimension of a code $\mathcal C\subseteq 2^{[n]}$. Closed-dimension existence conjecture. There exists a code $\mathcal C\subseteq 2^{[n]}$ such that

$$n<\operatorname{cdim}(\mathcal C)<\infty.$$

The paper notes that no such example was known there, despite analogous open embedding dimensions being able to exceed $n$; finding one would provide a starting point for understanding the minimum base set size needed for a prescribed closed embedding dimension.