Dimension bound and equality characterization for binary graphical descriptions

Fix a tuple of graphs $(\mathcal{G}_1,\dots,\mathcal{G}_d)$ on vertex set $[r]$, and let $W \subset (\mathbb{R}^2)^{\otimes d}$ be the set described by this tuple. Assume that the tuple is a valid graphical description, so each $\mathcal{G}_j$ is a disjoint union of $c_j$ complete bipartite graphs. Graphical-description dimension conjecture. Then

$$r+\sum_{j=1}^d c_j\leq 2^{d-1}+d,$$

with equality only if $r=2^{d-1}$. This proposed bound would constrain the expected dimensions of all valid graphical-description varieties and would identify the only possible equality value of the number of summands.