Limit conjecture for expected modularity above the Erdős–Rényi connectivity threshold

For a constant $c>0$, let $G_{n,c/n}$ be the binomial Erdős–Rényi random graph with edge-probability $c/n$, and define

$$\bar{q}(n,c)={\mathbb E}[q^*(G_{n,c/n})].$$

For $0<c\leq1$, the source states that $\bar{q}(n,c)\to1$ as $n\to\infty$ and sets $\bar{q}(c)=1$. Expected-modularity limit conjecture. For each $c>1$,

$$\bar{q}(n,c)\longrightarrow\bar{q}(c)\qquad\text{as }n\to\infty.$$

This is the unresolved part of the broader limit conjecture for expected modularity of sparse Erdős–Rényi graphs; continuity of the limiting function was noted conditional on existence of the limits.