We worked out the details to get what we needed in that paper. Specifically, if $f$ is an idempotent upper-semicontinuous continuum-valued function from $I$ to $I$ (equivalently, idempotent with a graph which is closed and connected satisfying $f(x)=[l(x),u(x)]$), then the graph of $f$ satisfies what we called condition $\Gamma$: there exists $x,y\in I$ such that $\langle x,x\rangle,\langle y,y\rangle,\langle x,y\rangle$ are all in the graph of $f$.
It would be interesting if continuum-valued was not necessary.
Edit: In https://arxiv.org/pdf/1805.06827.pdf we weakened the assumptions to any weakly countably compact space, and any idempotent relation that's a closed subset of the square and yields a non-empty image and pre-image for each point.