Jørgensen's Conjecture: is every 6-connected graph with no K_6 minor apex?

A graph $H$ is a *minor* of $G$ if $H$ can be obtained from a subgraph of $G$ by contracting edges. A graph is *apex* if deleting some single vertex leaves a planar graph. Jørgensen (1994) conjectured that **every $6$-connected graph with no $K_6$ minor is apex**. Decide the conjecture, or exhibit a $6$-connected $K_6$-minor-free graph that is not apex. Computational sub-question: verify the conjecture for all $6$-connected graphs up to $n$ vertices.

Acceptance. FULLY RESOLVES: a $6$-connected $K_6$-minor-free non-apex graph, with a certificate that it has no $K_6$ minor (a rooted-minor/branch-decomposition test) and that deleting any single vertex still leaves a non-planar graph (a planarity test per vertex), refuting the conjecture; or a proof. PARTIAL PROGRESS: an exhaustive verification, via $\texttt{nauty}$, that every $6$-connected graph on at most $n$ vertices with no $K_6$ minor is apex. Provide the minor test, per-vertex planarity check, and enumeration bound.