Conjectured sharp joint density bound for near-unit-circle roots

Let $\mathbb{T}_0$ be the domain of the ordered $k$-tuples used in the paper, and let $p_k(\mathbf{x},\mathbf{y})$ denote the corresponding joint density. Sharp density conjecture. For all $\mathbf{x},\mathbf{y}\in\mathbb{T}_0$,

$$p_k(\mathbf{x},\mathbf{y})=\Theta_k\left(n^{k^2+k}\left(\prod_{i<j}\min\{|x_i-x_j|,n^{-1}\}\cdot\min\{|y_i-y_j|,n^{-1}\}\right)\right).$$

The paper currently proves only an upper bound of this form and states that the sharper asymptotic is suspected to be correct; no proof or disproof is given.