Near-sharp sphere-counting conjecture

Let $S(n,d)$ denote the number of $(d+2)$-tuples of points in $[n]^d$ that lie in a $(d-1)$-dimensional sphere. Sphere-counting conjecture. For any integer $d\geqslant 2$,

$$S(n,d)=O\left(n^{d^2+d-1}\right).$$

This conjecture would give an upper bound close to the lower bound in the paper's sphere-counting theorem. The paper's current upper bound leaves a gap, so this stronger estimate remains open.