Common-basis conjecture for partitions of a matroid ground set

Let $k\geq 2$, and let $M$ be a matroid of rank $r$ on a ground set $E$ with $|E|=kr+1$. Suppose $x,y\in E$ are such that both $E\setminus\{x\}$ and $E\setminus\{y\}$ can be partitioned into $k$ pairwise disjoint bases.

Common-basis conjecture. There exist partitions of $E\setminus\{x\}$ and $E\setminus\{y\}$ into $k$ pairwise disjoint bases that share a common basis.

This is the second conjecture introduced as closely related to White's conjectures. It expresses a common-basis condition for two complementary decompositions and is not resolved in the supplied text.