Strict increase of maximal embedding dimensions with base set size

Let $[n]=\{1,\ldots,n\}$, and for a code $\mathcal C\subseteq 2^{[n]}$ write $\operatorname{odim}(\mathcal C)$, $\operatorname{cdim}(\mathcal C)$, and $\operatorname{ndim}(\mathcal C)$ for its open, closed, and non-degenerate embedding dimensions. For each $n$, consider the maximum finite embedding dimension attained among codes on $[n]$. Strict-increase conjecture. The maximum open embedding dimension is a strictly increasing function of $n$: if $\mathcal C\subseteq 2^{[n]}$ and $\mathcal D\subseteq 2^{[n+1]}$ attain the respective maxima, then

$$\operatorname{odim}(\mathcal D)>\operatorname{odim}(\mathcal C).$$

The analogous strict increase should hold for closed and non-degenerate embedding dimensions. This would describe how efficiently neural codes on larger base sets can realize high-dimensional structures; the paper does not establish the conjecture, and the corresponding maxima remain a subject for further study.