Let $B$ be the radius-$r$ ball and $B'$ the radius-$(r+1)$ ball of the $𝒬$-flip metric around $v$, and $T' = T ∩ B'$. The transversal has one vertex in each part of $𝒫$, so $|T'| ≤ |B'/𝒫| ≤ k$. Take $u ∈ B$ and $t ∈ T ∖ B'$. Some $𝒬$-flip $H$ has no short walk from $v$ to $t$, so $u$ and $t$ are distinct and non-adjacent in $H$. Hence the adjacency of $t$ in $G$ is the same for all vertices of $B$ in the part $𝒬(u)$, which contains $𝒫(u)$. So the part of $u$ in $𝒫 ∧ T$ is determined by its part in $𝒫$ and its trace on $T'$. There are at most $k$ parts of $𝒫$ meeting $B$, and at most $π_G(k)$ traces on $T'$.