Kawarabayashi–Maharry conjecture for 5-connected non-planar graphs

Let $G$ be a $5$-connected non-planar graph with at least seven vertices. Kawarabayashi–Maharry conjecture. The graph $G$ contains both a $K_{3,4}$ minor and a $K_6^-$ minor. The source presents this as a proposed strengthening of their earlier result for sufficiently large almost $5$-connected non-planar graphs; no resolution is given here.