3 problems

Exceptional-density conjecture. The density $\rho_g(a,d)$ exists and, for at least $\gg_d x$ integers $g$ satisfying $|g|\leq x$, one has

Coprime-order completeness conjecture. Then $m$ is complete.

Primitive-number conjecture. The number $m$ is square-free, and there exists an index $i$ such that