Planar-surface horoball-packing conjecture

Let $F$ be a planar surface admitting a hyperbolic structure. A horoball packing of $F$ is a collection of pairwise disjoint horoball neighborhoods of its cusps, and the area of a horoball is its area in the induced hyperbolic metric.

Planar-surface horoball-packing conjecture. For any horoball packing of $F$, there is at least one horoball of area less than

$$\frac{10}{\sqrt{3}}.$$

If true, this would imply that the constructions in the paper produce asymptotically the longest possible reducible fillings. The conjecture is presented as an upper-bound problem for maximal-density horoball packings on planar surfaces, and the source gives no resolution.