Hyperbolic analogue of the Bezdek–Connelly–Kertész average-degree conjecture

Let ${\mathcal P}_r$ be a packing of disks of radius $r>0$ in the hyperbolic plane $\mathbb{H}^{2}$, and let ${\rm \deg}_{\rm avr}({\mathcal P}_r)$ denote the average degree of its contact graph. Hyperbolic average-degree conjecture. For arbitrary such packings,

$$\limsup_{r\to 0}\left(\sup_{{\mathcal P}_r}{\rm \deg}_{\rm avr}({\mathcal P}_r)\right)<5.$$

This is the still-open hyperbolic counterpart of the corresponding spherical result, which is known.