Ding and Marshall's minor conjecture for 3-connected non-Hamiltonian graphs

Let $G$) be a 3-connected non-Hamiltonian graph. Let $K_{3,4}$ be the complete bipartite graph, let $\mathfrak{Q}^+$ be obtained from the cube by adding a new vertex adjacent to three independent vertices, and let $\mathfrak{H}$ denote the Herschel graph. Ding and Marshall's conjecture. Every $3$-connected non-Hamiltonian graph contains a minor of $K_{3,4}$, $\mathfrak{Q}^+$, or $\mathfrak{H}$. This conjecture concerns the characterization of minor-minimal 3-connected non-Hamiltonian graphs; the paper states that the analogous characterization problem for 4-connected graphs remains open.