Uniform affine Dimension Growth Conjecture for hypersurfaces

Let $H\subset\mathbb{A}^n$ be the hypersurface defined by a polynomial $F\in\mathbb{Z}[X_1,\ldots,X_n]$ of degree $d\geq 2$, with absolutely irreducible leading form. Let $N_{\mathrm{aff}}(H,B)$ denote the number of integral points on $H$ of height at most $B$. Uniform affine Dimension Growth Conjecture. One should have

$$N_{\mathrm{aff}}(H,B)\ll_{n,d}B^{\dim_{\mathbb{A}}H-1}s(B),$$

where $s(B)$ is a small factor, potentially satisfying $s(B)\ll_\varepsilon B^\varepsilon$ for every $\varepsilon>0$, or $s(B)\leq(\log B)^{\gamma(n,d)}$ for some $\gamma(n,d)>0$, or $s(B)\leq 1$. This isolates the affine hypersurface cases needed to transfer progress to projective dimension-growth questions.