Strong Hadwiger's conjecture

Let $G$ be a graph with chromatic number $[?]{}$ and let $S\subseteq V(G)$ be colorful if every proper $\chi(G)$-coloring of $G$ assigns all $\chi(G)$ colors to vertices in $S$. An $S$-rooted $K_t$-minor is a model $(B_i)_{i=1}^{t}$ of $K_t$ in $G$ such that $B_i\cap S\neq\emptyset$ for every $i\in[t]$. Strong Hadwiger's conjecture. If $G$ is a graph with $\chi(G)=t$ and $S$ is a colorful set in $G$, then $G$ contains an $S$-rooted $K_t$-minor. This strengthens Hadwiger's conjecture by requiring the complete minor to meet a prescribed set that is hard to avoid in every optimal coloring; its status is not resolved in the supplied source.