A bipartite degree-four theorem and an exact duality obstruction

The graph complement conjecture asks whether $\operatorname{mr}(G)+\operatorname{mr}(\overline G)\le n+2$ for every graph on $n$ vertices. Take a bipartite graph whose vertices on one side have degree at most $d$. Vectors $(1,t_i,\dots,t_i^d)$ at distinct positive nodes and the coefficient vectors of padded neighbourhood polynomials give a faithful nonzero Gram representation of its complement in dimension $d+1$. Complements of partial 2-trees have faithful representations in dimension four, built by an explicit induction. These constructions are combined with minor monotonicity of the positive semidefinite Strong Arnold parameter. This proves the stronger bound $\operatorname{mr}_+(G)+\operatorname{mr}_+(\overline G)\le n+2$ for every bipartite graph whose degrees on one side are at most four, with no restriction on the order or on the other side. Separately, for the path on four vertices an open set of rank-two objectives has a unique optimizer that loses a required edge. So a natural semidefinite-duality route does not automatically produce matrices with the exact support.