Rainbow base factorization conjecture for bounded partition classes

Let $M$ be a $k$-base matroid, meaning that its ground set can be partitioned into $k$ bases, and let $\mathcal{P}$ be a partition of $E(M)$ such that

$$|X|\leq k-1$$

for every $X\in\mathcal{P}$. Rainbow base factorization conjecture. Then $M$ can be factorized into $k$ rainbow bases. The case $k=2$ is stated to be trivial, while the conjecture is presented as an optimistic open problem motivated by factorization under small partition classes.