This is a follow-up on an older question.
Suppose $G$ is a finite simple graph and $\eta(G)$ is the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$. Let $\delta(G)$ is the minimal degree of $G$. As Fedor Petrov showed, we do not necessarily have $\delta(G) \leq \eta(G)$.
Question. Is there necessarily an induced subgraph $G_0$ of $G$ with $\eta(G_0) = \eta(G)$ and $\delta(G_0) \leq \eta(G_0)$?
