Franke–Muthiah conjecture on two-dimensional convex embeddings

Let $\mathcal{C}$ be a code. Its minimal convex embedding dimension is the least dimension in which it admits a convex realization, and its minimal open convex embedding dimension is the least dimension in which it admits an open convex realization. Franke–Muthiah's conjecture. If $\mathcal{C}$ is open convex and has minimal open convex embedding dimension $2$, then the minimal convex embedding dimension of $\mathcal{C}$ is $2$. This conjecture concerns the relation between open convex and convex realizations in dimension $2$; the source gives no resolution.