Total-variation convergence for the multineighbor construction

Let $\beta=k-\frac{1}{n_m}$, $p=n^{-\frac{1}{\beta}}$, and $q=n^{-\frac{km}{\beta}}$, with $n=n_m$. Let $\Gamma_{n,m,p}$ and $Y_{n,k,q}$ denote the two distributions defined in the paper. Total-variation convergence conjecture. For every $k$, there is a sequence $n_m$ such that the total variation distance between the distributions $\Gamma_{n,m,p}$ and $Y_{n,k,q}$ tends to zero as $m$ tends to infinity. The conjecture proposes an asymptotic equivalence of these two random-complex distributions for the specified parameter choice, but the source does not establish resolution.