Alspach–Mason–Pullman conjecture on path decompositions of even tournaments

Let $D$ be a directed graph with vertex set $V(D)$ and edge set $E(D)$. A path decomposition of $D$ is a collection of directed paths whose edge sets partition $E(D)$. Let ${\rm pn}(D)$ be the minimum number of paths in a path decomposition of $D$. For each vertex $v\in V(D)$, define its excess by ${\rm ex}(v)=d_D^+(v)-d_D^-(v)$, its positive excess by ${\rm ex}^+(v)=\max\{{\rm ex}(v),0\}$, and its negative excess by ${\rm ex}^-(v)=\max\{-{\rm ex}(v),0\}$. Define

$${\rm ex}(D)=\sum_{v\in V(D)}{\rm ex}^+(v)=\sum_{v\in V(D)}{\rm ex}^-(v)=\frac12\sum_{v\in V(D)}|{\rm ex}(v)|.$$

A tournament is an orientation of a complete graph, and it is consistent when ${\rm pn}(D)={\rm ex}(D)$. Alspach–Mason–Pullman conjecture. Every tournament $T$ with an even number of vertices satisfies

$${\rm pn}(T)={\rm ex}(T).$$

The excess gives a general lower bound for the number of paths in a path decomposition. The conjecture asserts that this lower bound is always attained for even tournaments; the supplied source gives no resolution status.