Expected codimension and irreducibility of Spohn CI varieties for Bayesian networks

Let $\mathcal{C}$ be a Bayesian network on $n$ binary random variables, let $\mathcal{M}_\mathcal{C}$ be its model in $\mathbb{P}^{2^n-1}$, and let $\mathcal{V}_{X,\mathcal{C}}$ be the associated Spohn CI variety.

Expected-codimension and irreducibility conjecture. The variety $\mathcal{V}_{X,\mathcal{C}}$ has expected codimension $n$ inside $\mathcal{M}_\mathcal{C}$. It is positive-dimensional and irreducible whenever the network has at least one edge.

These properties were verified for all models computed in the paper. The conjecture proposes that the observed expected codimension always holds and that irreducibility fails only for the network with no edges.