A sharp density bound for 5-connected graphs with no $\Ke$ minor

Let $\Ke$ be obtained from $K_7$ by deleting two independent edges. We prove that every 5-connected graph on $n\ge7$ vertices with at least $4n-9$ edges contains a $\Ke$ minor, settling Conjecture~1.4 of Dvo\v rák, Norin and Rahman (arXiv preprint 2609.17760v1). The bound is sharp. We prove the stro