  We prove that for all $r\in \mathbb{N}\cup \{0\}$ and $s,t\in \mathbb{N}$,
there exists $\Omega=\Omega(r,s,t)\in \mathbb{N}$ with the following property.
Let $G$ be a graph and let $H$ be a subgraph of $G$ isomorphic to a $(\leq
r)$-subdivision of $K_{\Omega}$. Then either $G$ contains $K_t$ or $K_{t,t}$ as
an induced subgraph, or there is an induced subgraph $J$ of $G$ isomorphic to a
proper $(\leq r)$-subdivision of $K_s$ such that every branch vertex of $J$ is
a branch vertex of $H$. This answers in the affirmative a question of Lozin and
Razgon. In fact, we show that both the branch vertices and the paths
corresponding to the subdivided edges between them can be preserved.
