Theta-order modularity conjecture for preferential attachment graphs

Let $G_n^h=(V,E)$ be a preferential attachment graph, where $n$ is the number of vertices and $h$ is the number of edges added per step. Write $\operatorname{mod}(G_n^h)$ for its modularity, and let “whp” mean with high probability. The theta-order modularity conjecture. With high probability,

$$\operatorname{mod}(G_n^h)=\Theta(1/\sqrt{h}).$$

This conjecture would sharpen the known lower bound on modularity to the correct order in $h$, substantially improving the available upper bound. The source explicitly states that it remains open.