Failure of moment matching for the multineighbor construction

Let $\Gamma_{n,m,p}$ and $Y_{n,k,q}$ be the distributions considered in the paper, let $z_Kx$ denote the relevant count associated with a positive $k$-pure shape $x$, and set $p=n^{-\frac{1}{\beta}}$ and $q=n^{-\frac{km}{\beta}}$. Moment-ratio conjecture. There is a choice of $m$, $\beta$, and a positive $k$-pure shape $x$ such that the support of $\Gamma_{n,m,p}$ is the same as that of $Y_{n,k}$, but

$$\lim_{n\rightarrow\infty}\frac{\mathbb{E}_{K\in\Gamma_{n,m,p}}(z_Kx)}{\mathbb{E}_{K\in Y_{n,k,q}}(z_Kx)}\not=1.$$

Thus, even when the two distributions have the same support, the associated expected counts need not be asymptotically equal.