Global rigidity conjecture for complete graphs in the infinity norm

Let $d\geq 1$ and let $K_{2d+1}$ be the complete graph on $2d+1$ vertices. A realisation of $K_{2d+1}$ in $\ell_\infty^d$ assigns a point of $\mathbb{R}^d$ to each vertex, and it is generic when its coordinates satisfy no nontrivial algebraic relations over the rationals. The realisation is globally rigid if every realisation in $\ell_\infty^d$ with the same edge lengths is congruent to it. Global rigidity conjecture. The complete graph $K_{2d+1}$ has a generic globally rigid realisation in $\ell_\infty^d$. This is known in dimension $d=2$, while the assertion is unclear in the remaining dimensions.