The non-closed-convexity conjecture of Goldrup and Phillipson

Let $\mathcal{C}$ be a max-intersection incomplete open convex code, where $\Delta(\mathcal{C})$ has at least two non-mandatory codewords not contained in $\mathcal C$. Suppose $\mathcal{C}$ has at least three maximal codewords $M_1, M_2, M_3$, and there is $\sigma\subset M_1$ with $\sigma\in \mathcal{C}$ such that $\sigma\cap M_2\not\in \mathcal{C}$. Goldrup–Phillipson's conjecture. Then $\mathcal{C}$ is not closed convex. The conjecture proposes a criterion distinguishing open-convex codes that are not closed-convex; the source presents it as a conjecture of Goldrup and Phillipson, but the supplied text does not establish its resolution.