The Goemans-Linial integrality gap for uniform sparsest cut is nearly square-root logarithmic

Theorem 1.1: there are an absolute $c>0$ and uniform-demand instances on $n_j\to\infty$ vertices with Goemans-Linial integrality gap at least $c\sqrt{\log n_j}/(\log\log n_j)^3$. This improves the Kane-Meka bound $\exp(\Omega(\sqrt{\log\log n}))$ to within $(\log\log n)^3$ of the ARV upper bound. It is an existential sequence of sizes, not a bound at every $n$; it does not determine whether the gap is exactly $\Theta(\sqrt{\log n})$, and it concerns the basic SDP, not Sherali-Adams or Lasserre strengthenings.

Theorem 1.1: there are an absolute $c>0$ and uniform-demand instances on $n_j\to\infty$ vertices with Goemans-Linial integrality gap at least $c\sqrt{\log n_j}/(\log\log n_j)^3$. This improves the Kane-Meka bound $\exp(\Omega(\sqrt{\log\log n}))$ to within $(\log\log n)^3$ of the ARV upper bound. It is an existential sequence of sizes, not a bound at every $n$; it does not determine whether the gap is exactly $\Theta(\sqrt{\log n})$, and it concerns the basic SDP, not Sherali-Adams or Lasserre strengthenings.