The Malle conjecture for 2-ranks of cyclic cubic and quintic class groups

Let $\ell\in\{3,5\}$ and put $q=2^{\ell-1}$. For $m\in\mathbb{Z}_{\geq0}\cup\{\infty\}$, define $(q)_0=1$ and $(q)_m=\prod_{i=1}^m(1-q^{-i})$ for positive finite $m$.

Malle's class-group rank conjecture. As $K$ ranges over cyclic number fields of degree $\ell$, for every $r\in\mathbb{Z}_{\geq0}$,

$$\operatorname{Prob}\left(\operatorname{rk}_2\operatorname{Cl}(K)=(\ell-1)r\right)=\left(1+\frac{1}{\sqrt q}\right)\frac{(\sqrt q)_\infty(q^2)_\infty}{(q)_\infty^2}\frac{1}{\sqrt q^{r(r+2)}(q)_r}.$$

This is attributed in the source to Malle and corrects the Cohen–Lenstra prediction to account for roots of unity in these fields. The source gives no resolution.