Near-square-root logarithmic integrality gaps for uniform sparsest cut

We construct uniform sparsest-cut instances whose Goemans–Linial semidefinite integrality gap is at least $c\sqrt{\log n}/(\log \log n{)}^3$ along a sequence $n\to \infty$. The demand is one between every pair of distinct vertices, and the capacities are nonnegative real numbers. This matches the Arora–Rao–Vazirani upper bound up to a power of $\log \log n$.