Path decompositions of tournaments

In 1976, Alspach, Mason, and Pullman conjectured that any tournament $T$ of even order can be decomposed into exactly ${rm ex}(T)$ paths, where ${rm ex}(T):= frac{1}{2}sum_{vin V(T)}|d_T^+(v)-d_T^-(v)|$. We prove this conjecture for all sufficiently large tournaments. We also prove an asymptotically optimal result for tournaments of odd order.