Unbounded open embedding dimension for Steiner triple codes

A Steiner triple system on $n$ is a set of triples in $[n]$ in which every pair appears in a unique triple; adjoining the singletons and the empty codeword gives an associated Steiner triple code. For a code $\mathcal{C}$, write $\operatorname{odim}(\mathcal{C})$ for its open embedding dimension.

Steiner triple code embedding-dimension conjecture. For every $d\ge 1$ there exists a Steiner triple code $\mathcal{C}$ with

$$\operatorname{odim}(\mathcal{C})\ge d.$$

Steiner triple codes are 3-sparse and have closed embedding dimension at most five, while the Fano-plane example shows that their open embedding dimension can exceed their closed embedding dimension. The conjecture predicts that this open embedding dimension is unbounded across Steiner triple codes.