Geelen's conjecture on preserving two matroid connectivities

For a matroid $M$ on ground set $E(M)$, let $Q,R,S,T\subseteq E(M)$ satisfy $Q\cap R=S\cap T=\emptyset$. Write

$$k:=\kappa_M(Q,R),\qquad l:=\kappa_M(S,T).$$

Let $c:\mathbb{N}^2\to\mathbb{N}$ be a function. Geelen's conjecture. There exists such a function $c$ with the following property: if

$$|E(M){-}(Q\cup R\cup S\cup T)|\geq c(k,l),$$

then some element $e\in E(M){-}(Q\cup R\cup S\cup T)$ satisfies either

$$\kappa_{M{-} e}(Q,R)=k\quad\text{and}\quad\kappa_{M{-} e}(S,T)=l,$$

or

$$\kappa_{M/e}(Q,R)=k\quad\text{and}\quad\kappa_{M/e}(S,T)=l.$$

Here $\kappa_M$ denotes the matroid connectivity between two disjoint sets. The conjecture predicts that a sufficiently large set of elements outside the four terminals contains an element whose deletion or contraction preserves both prescribed connectivities; the paper's abstract indicates that this is proved for sufficiently large representable matroids, with a stronger finite-field result, but the supplied material does not establish the conjecture in its full stated generality.