Pullman's conjecture on strongly consistent odd-regular graphs

Let $G$ be a finite undirected graph. An orientation of $G$ is obtained by directing each edge, and $G$ is strongly consistent if every orientation of $G$ is consistent, where consistency means that the minimum number of directed paths in an edge decomposition equals the excess

$$\operatorname{ex}(D)=\frac{1}{2}\sum_{v\in V(D)}\lvert d^+(v)-d^-(v)\rvert.$$

Pullman's conjecture. For odd $d$, every $d$-regular graph is strongly consistent. This generalizes the even-order tournament conjecture from complete graphs to all odd-regular graphs; the paper develops results for sparse graphs but does not resolve the conjecture in full.