Jørgensen's apex conjecture for 6-connected graphs without $K_6$ minors

Let $G$ be a graph, and recall that $G$ is 6-connected if deleting fewer than six vertices leaves it connected, while $G$ is apex if it has a vertex $v$ such that $G-v$ is planar. A $K_6$ minor is a graph obtained from $G$ by vertex deletions, edge deletions, and simple edge contractions.

Jørgensen's conjecture. If $G$ is 6-connected and does not have a $K_6$ minor, then $G$ is apex.

This conjecture, attributed in the source to Jørgensen, is related to Hadwiger's conjecture and concerns the structure of highly connected graphs excluding a complete-graph minor. The supplied text gives no resolution status.