  Hadwiger's famous coloring conjecture states that every $t$-chromatic graph
contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp.
139--144] conjectured the following strengthening of Hadwiger's conjecture: If
$G$ is a $t$-chromatic graph and $S \subseteq V(G)$ takes all colors in every
$t$-coloring of $G$, then $G$ contains a $K_t$-minor rooted at $S$. We prove
this conjecture in the first open case of $t=4$. Notably, our result also
directly implies a stronger version of Hadwiger's conjecture for $5$-chromatic
graphs as follows: Every $5$-chromatic graph contains a $K_5$-minor with a
singleton branch-set. In fact, in a $5$-vertex-critical graph we may specify
the singleton branch-set to be any vertex of the graph.
