The strengthened lower bound for non-hexagonal disk packings

Let $S$ be a subset of $\mathds{R}^2$ such that no two of its points are at distance less than $2$, let $S^*$ be the set of center-points in a hexagonal close-packing of disks of radius $1$, and define

$$\delta(S)=\liminf_{s\rightarrow\infty}\left(\sum_{(x,y)\in S^*}e^{-(x^2+y^2)/s^2}-\sum_{(x,y)\in S}e^{-(x^2+y^2)/s^2}\right).$$

Strengthened packing conjecture. In the conclusion of the hexagonal packing uniqueness conjecture, the assertion $\delta(S)>0$ can be replaced by the stronger assertion $\delta(S)\geq 1$ for sets $S$ not related to $S^*$ by an isometry of $\mathds{R}^2$. This strengthens the non-isometric case of the preceding conjecture; the source provides no resolution of this quantitative bound.