Projective Dimension Growth Conjecture

Let $V\subset\mathbb{P}^N$ be an irreducible projective variety of degree $d\geq 2$ defined over $\mathbb{Q}$, and let $N_{\mathrm{proj}}(V,B)$ denote the number of rational points on $V$ of height at most $B$. Projective Dimension Growth Conjecture. There should be a constant $C(V)$ such that

$$N_{\mathrm{proj}}(V,B)\leq C(V)B^{\dim_{\mathbb{P}}V}s(B)\qquad\text{for all $B\geq 1$,}$$

where $s(B)$ is a small factor, potentially satisfying $s(B)\ll_\varepsilon B^\varepsilon$ for every $\varepsilon>0$, or $s(B)\leq(\log B)^{\gamma(V)}$ for some $\gamma(V)>0$, or $s(B)\leq 1$. The exponent is essentially best possible, and examples show that a logarithmic factor can be necessary.