Bounded Markov complexity for independent subsets

Let $\Delta$ be a simplicial complex on vertices $\{1,\ldots,n\}$, let its underlying graph be the graph whose edges are the two-element faces of $\Delta$, and let $\mathcal{M}_{\Delta,d}$ denote the universal Markov basis for the hierarchical model associated with $\Delta$ and level vector $d=(d_1,\ldots,d_n)$. Suppose that $\{1,2,\ldots,j\}$ is an independent subset of the underlying graph of $\Delta$. For fixed $d_{j+1},\ldots,d_n$, there exist numbers $(m_1,\ldots,m_j)=m(\Delta;d_{j+1},\ldots,d_n)$ such that every element in $\mathcal{M}_{\Delta,d}$ has format smaller than

$$m_1\times\cdots\times m_j\times d_{j+1}\times\cdots\times d_n.$$

Bounded Markov complexity conjecture. If $\{1,2,\ldots,j\}$ is an independent subset of the underlying graph of $\Delta$, then the preceding bounded-format conclusion holds. This would generalize the established result for reducible simplicial complexes when the varying levels correspond to nonadjacent vertices. The conjecture asserts that independence alone is sufficient for bounded Markov complexity as the levels vary.