Cocycle Cheeger constants for Steinberg and GKKL presentations

Let $SL_n(\mathbb{F}_p)$ be presented either by the Steinberg presentations, whose complexity is $\Omega(p)$, or by the Guralnik–Kantor–Kassabov–Lubotzky (GKKL) presentations, whose complexity is logarithmic in the size of the group. Let the cocycle Cheeger constant in permutations refer to the constant associated with one of these presentations.

Cocycle-Cheeger conjecture. The cocycle Cheeger constants in permutations of the Steinberg presentations have a uniform positive lower bound, at least when $n$ is fixed and $p\to\infty$. In contrast, the cocycle Cheeger constants in permutations of the GKKL presentations tend to zero when either $n\to\infty$ or $p\to\infty$.

This proposes opposite asymptotic behavior for the expansive Steinberg presentations and the logarithmic-complexity GKKL presentations of finite simple groups; the source describes this case as completely open.