Prokhorenkova et al.'s modularity conjecture for preferential attachment graphs

Let $h\geq 2$ and let $G\sim G^h_n$ be the preferential attachment graph with parameter $h$. Write $q^*(G)$ for its modularity value.

Prokhorenkova et al.'s modularity conjecture. With high probability,

$$q^*(G)=\Theta(h^{-1/2}).$$

This conjecture predicts the asymptotic order of the modularity of $G^h_n$ as the parameter $h$ varies. The source attributes it to Prokhorenkova et al.; the paper establishes an upper bound bounded away from $1$ for fixed $h\geq2$, but does not resolve the conjectured order.