Tightness conjecture for the layer number of evenly distributed point sets

Let $\alpha>1$ be real and let $d\geq 3$. An $\alpha$-evenly distributed point set is a finite point set $X\subset\mathbb{R}^d$ satisfying the distribution condition defined in the paper, and $L(X)$ denotes its layer number.

Tightness conjecture. There exists an $\alpha$-evenly distributed point set $X\subset\mathbb{R}^d$ such that

$$L(X)\geq\Omega\left(|X|^{\frac{d+1}{2d}}\right).$$

This conjecture asserts that the upper bound $L(X)=O\left(|X|^{\frac{d+1}{2d}}\right)$ for higher dimensions is tight. The paper gives a construction establishing tightness in the planar case, but notes that constructing matching examples in dimensions $d\geq3$ is more difficult and suggests vector configurations as a possible approach.