Density Gap Conjecture for finite triangular-torus packings

Let $n$ be a positive integer. An integer is a triangular lattice number if it has the form

$$n_1^2+n_1n_2+n_2^2$$

for integers $n_1,n_2$. Let $\delta_{\Delta}$ denote the maximal density of the triangular packing, so that

$$\delta_{\Delta}=\frac{\pi}{\sqrt{12}}.$$

Density Gap Conjecture. If $n+1$ is a triangular lattice number but $n$ is not, then the most dense packing of $n$ equal disks in the triangular torus has density

$$\delta=\frac{n}{n+1}\delta_{\Delta}=\frac{n}{n+1}\frac{\pi}{\sqrt{12}},$$

and this density is achieved only by the triangular packing with one disk removed.

The conjecture gives a finite-torus formulation of the solidity question: the density gap would rule out rearrangements improving on the one-disk-deleted triangular packing. The paper states that it implies Fejes Toth's conjecture and provides evidence, but no proof.