The bounded-error conjecture for equitable matroid partitions

Let $\mathcal{M}=(E,\mathcal{I})$ be a matroid whose ground set $E$ can be partitioned into $k$ disjoint bases, and let $S_1,S_2,\ldots,S_\ell$ be pairwise disjoint subsets of $E$. Bounded-error equitable partition conjecture. There exists a function $f\,:\,\mathbb{N}\to\mathbb{N}$ such that there is a partition of $E$ into $k$ disjoint bases $B_1,\ldots,B_k$ satisfying

$$\left||B_i\cap S_j|-|S_j|/k\right|\le f(\ell)$$

for all $i\in [k]$ and $j\in[\ell]$. This conjecture asks whether the discrepancy can be bounded solely in terms of the number of distinguished subsets, independently of the matroid, $k$, and their sizes.