The dimension growth conjecture for smooth projective hypersurfaces

Let $n\geqslant 4$ and $d\geqslant 3$. Let $X$ be a smooth projective hypersurface defined over $\mathbb{Q}$ of degree $d$ in $\mathbb{P}^n$. Let $N(X,B)$ count the rational points on $X$ of height at most $B$.

Dimension growth conjecture. For every $\varepsilon>0$,

$$N(X,B)\ll_{n,d,\varepsilon} B^{n-2+\varepsilon}.$$

The paper states that this bound should hold for all $n\geqslant 4$ and $d\geqslant 3$. It is known in particular when $d\geqslant 50$ or $n\geqslant 28$, while the stated range remains open in general; the bound fails in general for $d\leqslant 2$ or $n\leqslant 3$.