Gowers' clique-difference conjecture for graphs

Let $\delta>0$. Consider the collection of non-oriented graphs on the vertex set $[n]$, with density measured relative to the number of such graphs. Gowers' clique-difference conjecture. For $n$ large enough depending on $\delta$, every subset of this collection with density at least $\delta$ contains distinct graphs $(H,G)$ such that $H$ is a subgraph of $G$ and the complement of $H$ inside $G$ is a clique. This is presented as a graph-theoretic conjecture related to the square-difference conjecture; the supplied text gives no resolution.