Heath-Brown's rational point density conjecture for hypersurfaces

Let $F\in\mathbb{Z}[x_1,\ldots,x_n]$ be an absolutely irreducible form of degree $d\geqslant 2$, and let $X_F\subset\mathbb{P}^{n-1}$ be the hypersurface defined by $F$. For $P\geqslant 1$, write

$$N_{X_F}(P)=\#\{x\in X_F\cap\mathbb{P}^{n-1}(\mathbb{Q}): H(x)\leqslant P\},$$

where $H(x)$ is the projective height. Heath-Brown's conjecture. For every $\varepsilon>0$,

$$N_{X_F}(P)=O_{d,\varepsilon,n}(P^{n-2+\varepsilon}).$$

This conjecture predicts the expected upper bound for rational points of bounded height on an absolutely irreducible hypersurface, uniformly in the degree, number of variables and error parameter. The source does not state whether the conjecture is resolved in this generality.