Suppose $$X$$, $$Y$$, and $$Z$$ are discrete distributions supported on $$1,2,\\dots,n$$.  We can ask about $$P(X<Y)$$ and $$P(Y<Z)$$ and $$P(X<Z)$$, which is what the Wilcoxon/Mann-Whitney rank test does.
The project has basically proved that under one model for randomly choosing distributions, if $$X$$,  $$Y$$, and $$Z$$ have the same mean and $$P(X>Y)>1/2$$ and $$P(Y>Z)>1/2$$, the probability of $$P(X>Z)>1/2$$ is $$1/2+o(1)$$. That is, if three distributions have the same mean, and the Wilcoxon test says $$X$$ is bigger than $$Y$$ and $$Y$$ is bigger than $$Z$$, you’ve got essentially no information about whether it will say $$X$$ is bigger than $$Z$$.
Gowers also says they are close to showing a converse: if the means are different, then $$P(X>Y)>1/2$$, $$P(Y>Z)>1/2$$ and $$P(X>Z)>1/2$$ are true or false they way you’d assume from the ordering of the means.