Lower-bound conjecture for the Cheeger constant of random Cayley graphs

There is a random Cayley graph on a finite group $G$ generated by $k$ generators. For every $\varepsilon\in(0,1)$, consider groups with sufficiently large size and take sufficiently many generators.

Cheeger lower-bound conjecture. There exists an absolute constant $c$ such that, for all $\varepsilon\in(0,1)$, there are constants $N(\varepsilon)$ and $M(\varepsilon)$ such that, for every finite group $G$ with $|G|\geq N(\varepsilon)$ and every $k\geq M(\varepsilon)$, the probability that the Cheeger constant is less than $c|G|^{-1/k}$ is at most $\varepsilon$.

This is a probabilistic lower-bound formulation of the claimed order $|G|^{-1/k}$ for the Cheeger constant. The supplied text does not provide evidence that this stronger formulation has been resolved.