Constant-q asymptotic limit-shape and discrete-sine-kernel conjecture

Consider the specializations

$$x_i=q^{i-1},\qquad y_j=q^{b(j-1)}.$$

Let $\mu_{n,k}^b$ be the corresponding measure, with $q>0$, $b\neq 0$, and let $\rho(t)$ and $\Omega(t)$ denote its limiting density and limit shape. Constant-q asymptotic conjecture. For $b<0$, the limiting density is $\rho(t)=\frac{b}{b-1}$ and the limit shape is the stated straight line, with the stated endpoint depending on whether $q>1$ or $q<1$; for $b>0$, the limit shape is empty when $q<1$ and full when $q>1$. Moreover, in the support interval the correlation kernel converges to the discrete sine kernel:

$$\lim_{n,k\to\infty}\mathcal{K}(nt+l,nt+l')=\begin{cases}\dfrac{\sin\bigl(\pi\rho(t)(l-l')\bigr)}{\pi(l-l')},&l\ne l',\rho(t),&l=l'.\end{cases}$$

This conjecture describes the degenerate constant-$q$ regime, including its linear or empty/full limit shapes and bulk sine-kernel fluctuations.