Projective type II thin-set counting conjecture

Let $M\subset\mathbb{P}^{n-1}(\mathbb{Q})$ be a thin set of type II, and let $N_{\mathbb{P}^{n-1}}(M,B)$ denote the number of points of $M$ of height at most $B$. Projective type II conjecture. There should be a constant $C(M)$ such that

$$N_{\mathbb{P}^{n-1}}(M,B)\leq C(M)B^{n-1}s(B),$$

where $s(B)$ is a small factor, potentially satisfying $s(B)\ll_\varepsilon B^\varepsilon$ for every $\varepsilon>0$, or $s(B)\leq(\log B)^{\gamma(M)}$ for some $\gamma(M)>0$, or $s(B)\leq 1$. This is the central conjecture for projective thin sets of type II and would sharpen the general baseline bound.