Double-critical graphs and complete minors

A connected $k$-chromatic graph $G$ is double-critical if for all edges $uv$ of $G$ the graph $G-u-v$ is $(k-2)$-colourable. The only known double-critical $k$-chromatic graph is the complete $k$-graph $K_k$. The conjecture that there are no other double-critical graphs is a special case of a conjecture from 1966, due to Erd\H{o}s and Lov'asz. The conjecture has been verified for $k\le 5$. We prove for $k=6$ and $k=7$ that any non-complete double-critical $k$-chromatic graph is 6-connected and has $K_k$ as a minor.