The EF1 conjecture for individually constrained matroid bundles

Let $M_1,\ldots,M_n$ be matroids on the same finite ground set $E$, with $\chi(M_i)\leq n$ for every $i=1,\ldots,n$. Suppose $n$ agents have additive valuations $v_1,\ldots,v_n$ of the goods or chores in $E$. The individually constrained matroid EF1 conjecture. There exists an EF1 partition

$$E=X_1\cup\cdots\cup X_n$$

such that $X_i$ is independent in $M_i$ for every $i=1,\ldots,n$. The source presents this as a general extension to individual matroid constraints; feasibility without EF1 is known, but the EF1 assertion is left as a conjecture.