The matroid intersection conjecture

We prove the matroid intersection conjecture, due to Nash-Williams \cite{ahaziv}, for infinite matroids: If $M_k = (E, {\mathcal I}_k)$ are matroids for $k \in \{0,1\}$, both having the same base set $E$, then there is $J^* \in {\mathcal I}_0 \cap {\mathcal I}_1$ and disjoint subsets $J_k\subseteq J^*$ for $k\in \{0,1\}$ such that $\text{cl}_{M_0}(J_0) \cup \text{cl}_{M_1}(J_1) = E$.