Browning–Heath-Brown conjecture on rational points of non-singular hypersurfaces

Let $n\geqslant3$, let $F\in\mathbb{Z}[X_0,\ldots,X_n]$ be a non-singular form of degree $d\geqslant2$, and define

$$N(F;B)=\#\{\mathbf{x}\in\mathbb{Z}^{n+1}:F(\mathbf{x})=0,\ \mathrm{h.c.f.}(x_0,\ldots,x_n)=1,\ |\mathbf{x}|\leqslant B\}.$$

For $B\geqslant1$, $|\mathbf{x}|=\max_{0\leqslant i\leqslant n}|x_i|$. Browning–Heath-Brown conjecture. For every $\varepsilon>0$,

$$N(F;B)=O_{d,\varepsilon,n}(B^{n-1+\varepsilon}).$$

This is presented as the paper's basic conjecture and is a special case of the stronger conjecture allowing $F$ to be merely absolutely irreducible; the source supplies no evidence of resolution.