Rainbow base factorization conjecture for 2-bounded partitions

Let $M$ be a $k$-base matroid with $k\geq 3$, and let $\mathcal{P}$ be a partition of $E(M)$ satisfying

$$|X|\leq 2$$

for every $X\in\mathcal{P}$. 2-bounded rainbow factorization conjecture. Then $M$ can be factorized into $k$ rainbow bases. This is presented as the first nontrivial special case of the broader bounded-partition conjecture; the source leaves it open.