Erdős problem 150

A minimal cut of a graph is a minimal set of vertices whose removal disconnects the graph. Let $c(n)$ be the maximum number of minimal cuts a graph on $n$ vertices can have.

Does $c(n)^{1/n}\to \alpha$ for some $\alpha <2$?

It is unclear in [Er88] whether Erdős knew that the limit existed, which follows from a simple argument first given in the literature (to the best of my knowledge) by Bradač [Br24].

That $\alpha<2$ was proved by Fomin, Kratsch, Todinca, and Villanger [FKTV08], who proved $\alpha \leq 1.7087$. This was independently studied by Bradač [Br24] (unaware of this earlier work), who proved that $\alpha \leq 2^{H(1/3)}\approx 1.8899$, where $H(\cdot)$ is the binary entropy function.