As in the paper, for each pair of parts of $𝒬 = 𝒫 ∧ T$ we find a flip of $G$ on these parts whose edges lie in $R$. Then the pairs outside $R$ are non-edges of the flip, so they are all edges or all non-edges of $G$. Two parts of $𝒬$ inside one part $X$ of $𝒫$: by the diameter lemma, $G[X]$ or Ḡ[X] has diameter at most $3$. In the first case every non-adjacent pair of $X$ is joined by a walk of length at most $4$ between opposite copies in $G[X, X]$ and by an edge in Ḡ[X, X], so it lies in $R$, and the second case is symmetric. Parts inside different parts $X$, $Y$ of $𝒫$: apply the bipartite flip lemma to $G[X, Y]$ with the representatives in $T$. The parts of $𝒬$ refine the four parts of that lemma, so the flip is constant on the pair.