Wooley's lower-bound conjecture for rational points on cubic hypersurfaces

Let $n\geqslant 10$ be an integer, let $X\subset \mathbf{P}_{\mathbf{Q}}^{n-1}$ be a cubic hypersurface, and let $N(X,B)$ denote the number of rational points on $X$ counted with the height bound used in the source. Thus $N(X,B)$ measures the distribution of rational points of $X$ up to height $B$.

Wooley's conjecture. There exists a constant $c>0$ depending only on $X$ such that

$$N(X,B)\geqslant cB^{n-9}$$

as $B\to\infty$.

This is a quantitative strengthening of the folklore local-to-global conjecture and is motivated by examples with substantially fewer points than the probabilistic heuristic predicts. The source attributes the conjecture to T. D. Wooley; its status is not resolved in the supplied text.