Bachoc–Robins conjecture on optimal density for tiling norms

Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$ whose unit ball tiles $\mathbb{R}^n$ by translation. Let $m_1(\mathbb{R}^n,\|\cdot\|)$ denote the supremum of the upper densities of measurable subsets of $\mathbb{R}^n$ avoiding distance $1$. Bachoc–Robins conjecture.

$$m_1(\mathbb{R}^n,\|\cdot\|)=\frac{1}{2^n}.$$

The construction from a translational tiling gives a distance-1-avoiding set of density $2^{-n}$, and the conjecture asserts that this construction is optimal. The source does not report a resolution.