Rainbow basis covering conjecture for 2-bounded partitions

Let $M$ be a $k$-base matroid, and let $\mathcal{P}$ be a 2-bounded partition of $E(M)$, meaning that every partition class has size at most $2$. Rainbow basis covering conjecture. There is an integer $k$ such that $M$ can be covered by $k$ bases, one of which is rainbow with respect to $\mathcal{P}$. This is described as a weaker possible first step toward the two rainbow factorization conjectures and is left open.