The CM Galois group realization conjecture for Weil numbers

Let $W_{2g}=C_2^g\rtimes S_g$ be the wreath product acting on the $2g$ conjugates in $g$ complex-conjugate pairs, and let $G$ be a transitive subgroup of $W_{2g}$ containing complex conjugation. A CM Galois group realization conjecture asserts that there is a Weil $q$-number $\pi$ such that

$$\operatorname{Gal}(\mathbf{Q}(\pi)^{\operatorname{gal}}/\mathbf{Q})\cong G.$$

The preceding lemma gives the necessary constraints on the Galois group in the usual case $d=g$; the conjecture says these constraints are also sufficient. The condition that $G$ contain complex conjugation is necessary, since a transitive subgroup of $W_8$ without complex conjugation does not arise as the Galois group of a CM-field.