Expected second-degree distribution in the Buckley–Osthus model with multiple edges

Let $a$ be the model parameter, let $m>1$, and let $H_{a,m}^n$ be the corresponding Buckley–Osthus random graph. For an integer $k$, let $Y_n^m(k)$ denote the number of $k$-vertices in $H_{a,m}^n$.

Expected second-degree distribution conjecture. For

$$k=O\left(n^{\min\left\{\frac{1}{2+a},\frac{1}{2a}\right\}}\right),$$

we have

$$\mathsf{E}Y_n^m(k)=\Theta\left(\frac{n}{k^a}\right).$$

This conjecture would extend the paper’s results on the distribution of second degrees from the case of one edge added at each step to arbitrary $m>1$. The paper explains that proving the needed analogue of an earlier result for $m>1$ requires additional calculations; no resolution is given here.