Strong-field convergence of hyperbolic to planar beta-exponent packing density

For positive real parameters $\alpha$ and $\beta$, let $\rho_{\alpha,\beta}(\mathbb{H})$ be the minimal discrepancy density for hyperbolic zero packing with field strength $\alpha$ and exponent $\beta$, defined by

$$\rho_{\alpha,\beta}(\mathbb{H})=\liminf_{r\to1^-}\inf_f\frac{1}{\log\frac{1}{1-r^2}}\int_{\mathbb{D}(0,r)}\big((1-|z|^2)^\alpha|f(z)|^\beta-1\big)^2\frac{\mathrm{d} A(z)}{1-|z|^2}.$$

Let $\rho_\beta(\mathbb{C})$ be the corresponding planar density. Strong-field convergence conjecture. We believe that

$$\lim_{\alpha\to+\infty}\rho_{\alpha,\beta}(\mathbb{H})=\rho_\beta(\mathbb{C}).$$

This expresses the expectation that only local effects matter as the field strength increases; the source gives no resolution.