  In $1980$ White conjectured that every element of the toric ideal of a
matroid is generated by quadratic binomials corresponding to symmetric
exchanges. We prove White's conjecture for high degrees with respect to the
rank. This extends our result arXiv:1302.5236 confirming White's conjecture `up
to saturation'. Furthermore, we study degrees of Gr\"{o}bner bases and Betti
tables of the toric ideals of matroids of a fixed rank.
