Kertész's density conjecture for TS-packings

Let ${\bf Q}$ be a $d$-dimensional cube containing a TS-packing of $N$ unit balls in ${\mathbb E}^d$, where $d\geq 4$. Write $\omega_d={\rm vol}_d\left({\mathbf B}^d[\mathbf{o},1]\right)$ for the volume of the unit ball, and let $\delta_d^{TS}$ denote the supremum of the upper densities of TS-packings of unit balls.

Kertész's density conjecture. If ${\bf Q}$ contains a TS-packing of $N$ unit balls, then

$${\rm vol}_d({\bf Q})\geq N2^d,$$

which implies

$$\delta_d^{TS}=2^{-d}\omega_d.$$

Here $2^{-d}\omega_d$ is the density of the TS-packing of unit diameter balls centered at the points of the integer lattice ${\mathbb Z}^d$ in ${\mathbb E}^d$.

This conjecture extends Kertész's theorem from dimensions two and three to higher dimensions. If true, together with the lower bound $\delta_d\geq\Omega(d2^{-d})$ and $\lim_{d\to\infty}\omega_d=0$, it would show that TS-packings have strictly smaller density than general, and hence locally separable, packings in all sufficiently large dimensions.