Primitive-point asymptotic for quotients of products of projective spaces

Let $G\subset S_n$ be a transitive subgroup, let $X=(\mathbb{P}^m)^n/G$, and assume $m\cdot\operatorname{ind}(G)\ge 2$. Let $X(K)^{\operatorname{prim}}$ denote the primitive $K$-points, obtained from the unramified locus by removing the images arising from proper subgroups of $G$. Primitive-point conjecture. There exists a positive constant $c$ such that

$$N_{X(K)^{\operatorname{prim}}}(B)\sim c\cdot B\cdot(\log B)^{\operatorname{cd}(X)}\qquad(B\to\infty).$$

This is a more specific version of the paper's Manin-type conjecture, designed to count points after removing a thin accumulating subset. Its validity is presented as conjectural and is used to study connections with Malle's conjecture.