Geelen's weak vertex-minor structure conjecture

Let $d53d$ be a proper vertex-minor-closed class of graphs. For $k,p displaymath \in \mathbb{N}$, a graph is $k$-rank-connected if it has at least $2k$ vertices and satisfies

$$\rho(X)\geq \min\bigl(|X|,|V(G)-X|,k\bigr)$$

for every vertex set $X$, where $\rho(X)$ is the rank over $GF(2)$ of the adjacency submatrix between $X$ and $V(G)-X$. A graph is a rank-$p$ perturbation of a circle graph if it can be obtained from a circle graph by a perturbation of rank at most $p$. Geelen's weak vertex-minor structure conjecture. There exist $k,p\in\mathbb{N}$ such that every $k$-rank-connected graph in $\mathcal{F}$ is a rank-$p$ perturbation of a circle graph. This conjecture would provide the structural reduction underlying the paper's bounds for proper vertex-minor-closed classes; the source gives no resolution.