Codimension conjecture for Spohn CI varieties of binary-choice games

Let $G$ be the undirected graphical model modelling a generic $n$-player game $X$ with binary choices in normal form. Let $\mathcal{C}=\operatorname{global}(G)$ and let $\mathcal{M}_{\mathcal{C}}$ be the discrete conditional independence model of $G$. Let $\mathcal{V}_{X,\mathcal{C}}$ be the corresponding Spohn conditional-independence variety. Codimension conjecture. The variety $\mathcal{V}_{X,\mathcal{C}}$ has codimension $n$ in $\mathcal{M}_{\mathcal{C}}$. The conjecture predicts a uniform codimension for the Spohn conditional-independence variety associated with a graphical model of a generic binary-choice game; the supplied text gives no resolution status.