Critical maximum-cut conjecture for random triangle-free graphs

Let $\mathcal{T}(n,m)$ be the set of triangle-free graphs on $n$ vertices with $m$ edges, and let a graph drawn uniformly from $\mathcal{T}(n,m)$ be denoted by $G$. For a graph $G$, a maximum cut is a cut containing the largest possible number of edges. Write $\operatorname{whp}$ for “with high probability” and $m\sim c n^{3/2}$ for an edge count asymptotic to $c n^{3/2}$.

Critical maximum-cut conjecture. There exists $c^*>0$ and a continuous function $\delta:(c^*,\infty)\to(0,1/2]$ such that:

$$\text{if }c<c^*\text{ and }m\sim c n^{3/2},\text{ then whp }G\text{ has a maximum cut of size }(1/2+o(1))m;$$

and

$$\text{if }c>c^*\text{ and }m\sim c n^{3/2},\text{ then whp }G\text{ has a maximum cut of size }(1/2+\delta(c)+o(1))m.$$

This gives a precise proposed order–disorder transition at the scale $m\sim n^{3/2}$: below the critical constant the maximum-cut fraction tends to $1/2$, while above it the fraction is separated from $1/2$ by the positive quantity $\delta(c)$. The source presents this as a conjecture and does not provide a resolution.