Matroid Intersection for Two Countable Nearly Finitary Matroids

We prove that if $ M $ and $ N $ are nearly finitary matroids on a common countable edge set $ E $ then they admit a common independent set $I $ such that there is a bipartition $ E=E_{M}\cup E_{N} $ for which $ I\cap E_M $ spans $ E_M $ in $ M $ and $ I\cap E_N $ spans $ E_N $ in $ N $. It answers positively the original form of the Matroid Intersection Conjecture of Nash-Williams in the countable case improving the partial result obtained by Aharoni and Ziv. However the problem for more general matroids remains open.