The dimension growth conjecture for rational points on projective varieties

Let $X\subseteq \mathbb{P}_{\mathbb{Q}}^{n}$ be an integral projective variety of degree $d\geq 2$. Let $H$ be the absolute projective multiplicative height. Dimension growth conjecture. For any $\varepsilon>0$ it holds:

$$|\{\boldsymbol x\in X(\mathbb{Q}):H(\boldsymbol x)\leq B\}|\lesssim_{\dim(X),d,\varepsilon}B^{\dim(X)+\varepsilon}.$$

This conjecture is a uniform form of the expected dimension-growth bound for rational points, generalising results of Bombieri–Pila and Heath-Brown. Its resolution is not indicated in the supplied context.