Browning–Heath-Brown–Salberger's rational-point conjecture for integral varieties

Let $Z\subset\mathbb{P}^N$ be an integral variety of degree $d\geq 2$ and dimension $m$. For a rational point $x=[\mathbf{x}]\in\mathbb{P}^N(\mathbb{Q})$, represented by a primitive integer vector, let $H(x)=|\mathbf{x}|$, and define

$$N_Z(B)=\#\{x\in Z\cap\mathbb{P}^N(\mathbb{Q}):H(x)\leq B\}.$$

Browning–Heath-Brown–Salberger's conjecture. For every $\varepsilon>0$,

$$N_Z(B)=O_{d,\varepsilon,N}(B^{m+\varepsilon}).$$

This generalizes the hypersurface conjecture to arbitrary integral projective varieties and predicts a uniform bound of essentially the dimension-sized exponent. The paper formulates it as an extension of the preceding conjecture; its resolution status is not specified in the supplied text.