Packing conjecture for graphs with many large complete bipartite subgraphs

Let $G$ be a graph on $N$ vertices. Suppose there is a parameter $W$ such that, for every induced subgraph $G'$ and every $n$, all but at most

$$\frac{W}{n}|V(G')|^2$$

edges of $G'$ are contained in some complete bipartite graph $K_{|V(G')|/n,|V(G')|/n}$. Say that two copies of $G$ pack into $K_N$ when they can be embedded on the same $N$-vertex set with edge-disjoint edge sets. Packing conjecture. If, in addition, two copies of $G$ pack into $K_N$, then

$$N=W^{O(1)}.$$

This generalizes the preceding Kneser-graph packing question by proposing that the abundance of large complete bipartite subgraphs obstructs packing unless the number of vertices is polynomially bounded in the exceptional-edge parameter. No resolution is given in the source.