Equality of the tight hyperbolic and hyperbolic minimal discrepancy densities

Let $\rho_{\mathbb{H}}$ be the minimal discrepancy density for hyperbolic zero packing, and let $\rho_{\mathbb{H}}^\ast$ be the minimal discrepancy density for tight hyperbolic zero packing, defined by

$$\rho_{\mathbb{H}}^\ast=\liminf_{r\to1^-}\inf_f\frac{\int_{\mathbb{D}}\Phi_f(z,r)\frac{\mathrm{d} A(z)}{1-|z|^2}}{\log\frac{1}{1-r^2}},$$

where $\Phi_f(z,r)=\big((1-|z|^2)|f(z)|-1_{\mathbb{D}(0,r)}(z)\big)^2$. Equality conjecture. We believe that $\rho_{\mathbb{H}}^\ast=\rho_{\mathbb{H}}$. This would identify the optimal asymptotic discrepancy for tight hyperbolic zero packing with the usual hyperbolic minimal discrepancy density; the source provides no resolution.