The conjecture on densest congruent geodesic ball packings in Thurston geometries

Let $X$ be a Thurston geometry, and let $\mathcal{B}$ be an arbitrary congruent geodesic ball packing in $X$ generated by a discrete isometry group of $X$. Let $\mathcal{B}_{\mathrm{opt}}(R_4,K_4)$ be the previously determined ball arrangement, whose density is $\delta(R_4,K_4)\approx 0.87757183$.

Densest-arrangement conjecture. The ball arrangement $\mathcal{B}_{\mathrm{opt}}(R_4,K_4)$ provides the densest congruent geodesic ball packing for the Thurston geometries.

The conjecture is motivated by the computed optimal arrangement in the multiply transitive $\boldsymbol{S}^2\times\boldsymbol{R}$ space-group example. The source notes that a general definition of density for congruent geodesic ball packings in Thurston geometries is not yet settled, so the claim remains open and depends on an appropriate definition of density.