Packing/covering conjecture for infinite matroid families

Let $(M_k\mid k\in K)$ be a family of matroids on the same ground set $E$. A packing consists of pairwise disjoint spanning sets, one for each $M_k$, and a covering consists of independent sets $I_k$ in $M_k$ whose union covers $E$. For $P\subseteq E$, write $M_k\restriction_P$ for restriction, and for $C\subseteq E$, write $M_k.C$ for contraction. Packing/covering conjecture. The ground set admits a partition

$$E=P\mathbin{\dot\cup}C$$

such that $(M_k\restriction_P\mid k\in K)$ has a packing and $(M_k.C\mid k\in K)$ has a covering. This conjecture is stated as equivalent to the matroid intersection conjecture and unifies the infinite base packing and base covering problems.