Conjecture on dimension two for open convex codes

Let $\mathcal{C}$ be a convex open code, meaning a code realizable by open convex sets. Its minimal convex embedding dimension is the smallest dimension in which it is realizable by convex sets, and its minimal open convex embedding dimension is the smallest dimension in which it is realizable by open convex sets.

Dimension-two embedding conjecture. If $\mathcal{C}$ has minimal open convex embedding dimension $2$, then its minimal convex embedding dimension is $2$.

This is stated as equivalent to the dimension-one conjecture above. It would show that, for open convex codes whose least open realization dimension is two, allowing non-open convex realizations does not reduce the dimension.