Malle's relative class-group distribution conjecture with roots of unity

Let $\ell$ be a prime, let $\mathcal{G}$ denote the set of finite abelian $\ell$-groups, and suppose that $A\in\mathcal{G}$ has $\ell$-rank $r$. Let $K_0$ be a number field containing $\ell$th roots of unity but not $\ell^2$th roots of unity. Let $\mathcal{S}$ be the set of quadratic extensions $K/K_0$ with a fixed signature and fixed relative unit rank $u$. For a relative class group, write $\operatorname{cl}(K/K_0)$ for the kernel of the norm map on ideal class groups, and write $\operatorname{cl}(K/K_0)[\ell^{\infty}]$ for its $\ell$-primary subgroup. Malle's relative class-group distribution conjecture. The limiting proportion of fields in $\mathcal{S}$ whose relative $\ell$-primary class group is isomorphic to $A$ should satisfy

$$\lim_{X\to\infty} \frac{\left|\left\{K\in\mathcal{S}\mid |\operatorname{Disc}K|\leq X,\ \operatorname{cl}(K/K_0)[\ell^{\infty}]\simeq A\right\}\right|} {\left|\left\{K\in\mathcal{S}\mid |\operatorname{Disc}K|\leq X\right\}\right|} = \frac{\prod_{i=u+1}^{u+r}(\ell^i-1)}{\ell^{r(u+1)}|A|^u|\operatorname{Aut}A|} \prod_{i=u+1}^{\infty}(1+\ell^{-i})^{-1}.$$

This is presented as a special case of Malle's proposed distributions for relative class groups when the base field contains $\ell$th but not $\ell^2$th roots of unity. The conjecture is intended to explain computational rank statistics that differ from the Cohen–Lenstra–Martinet predictions.