Strengthened vanishing correlation conjecture for lattice points near circles

Let $\mathcal{A}_R(\lambda_j)$ denote the normalized quantities attached to the cyclically ordered lattice points, and let $\mathcal{A}_\infty(r,\theta)$ be their limiting random model. For $k\geq1$, consider the random vectors

$$\left\{\left(\mathcal{A}_R(\lambda_j),\mathcal{A}_R(\lambda_{j+k})\right)\right\}_{1\leq j\leq K}.$$

Strengthened vanishing correlation conjecture. These vectors converge in distribution, in the double limit $R\to\infty$ followed by $k\to\infty$, to

$$(Y_1,Y_2):=\left(\mathcal{A}_\infty(r,\theta),\mathcal{A}_\infty(r',\theta)\right),$$

where $(r,\theta,r')$ is uniformly distributed on

$$G\times\left[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right] =\left[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right]\times[0,2\pi)\times\left[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right].$$

This is a strengthened form of the vanishing correlation conjecture: the preceding theorem identifies the limiting marginal distribution, while the conjecture asserts the stated joint limiting behavior as the separation $k$ tends to infinity. Its status is open in the supplied source.