Convergence of the graphical-model support approximation sequence

Let $G$ be an undirected graph with $d$ vertices. Let $\mathcal{X}\subseteq\mathbb{R}^d$ be an i.i.d. sample from a non-degenerate probability density $f_0$ on $\mathbb{R}^d$. Let $\mathcal{S}_{G,\mathcal{X}}$ be the support of the MLE $\hat{f}$ over $\mathcal{F}_G$, and let $\mathcal{D}^{(i)}_G$ be the sets defined in the paper.

Support approximation conjecture. The sequence of sets converges to the MLE support:

$$\lim_{i\to\infty}\mathcal{D}^{(i)}_G=\mathcal{S}_{G,\mathcal{X}}.$$

For non-chordal graphs, computing this limit is described as an open problem, while numerical experiments for a four-cycle provide evidence for the asserted convergence of the sets and their volumes.