The no-(d+2)-on-a-sphere conjecture for lattice cubes

Let $d \geq 3$ be a positive integer. A subset of the $d$-dimensional lattice cube $[n]^d$ is said to be admissible if no $d+2$ of its members lie on a sphere or a hyperplane.

No-$(d+2)$-on-a-sphere conjecture. There is an admissible subset of $[n]^d$ with

$$\Omega\left(n^{\frac{d}{d+1}}\right)$$

points.

The theorem preceding this conjecture gives a construction with $n^{\frac{3}{d+1}-o(1)}$ points. The conjecture proposes a stronger lower bound, while the authors note that an even stronger bound of order $\Omega(n)$ may be possible.