Conjecture on equality of minimal convex and open convex embedding dimensions in dimension one

Let $\mathcal{C}$ be a binary code. Its minimal convex embedding dimension is the smallest dimension in which it has a realization by convex sets. If $\mathcal{C}$ is open convex, its minimal open convex embedding dimension is the smallest dimension $k$ in which it is realizable by open convex sets in $\mathbb{R}^k$.

Dimension-one embedding conjecture. A code $\mathcal{C}$ has minimal convex embedding dimension $1$ if and only if it has minimal open convex embedding dimension $1$.

This is presented as a conjectural strengthening of the characterization by Rosen and Zhang of codes realizable by open convex sets in dimension one. It asserts that, in dimension one, allowing arbitrary convex sets does not lower the minimal realization dimension.