Cohen-Martinet density conjecture for class numbers of cyclic cubic fields

Let $K$ be a cyclic cubic number field and let $m$ be an integer not divisible by $3$. For a set of number fields, write $\mathrm{Prob}$ for its density, and let $h_K$ be the class number. Define

$$(p)_\infty=\prod_{k\geq 1}(1-p^{-k}),\qquad (p)_1=1-p^{-1}.$$

Cohen-Martinet conjecture.

$$\mathrm{Prob}(m\mid h_K)=\prod_{\substack{p\mid m\p\equiv 1\pmod 3}}\left(1-\frac{(p)_\infty^2}{(p)_1^2}\right)\prod_{\substack{p\mid m\p\equiv 2\pmod 3}}\left(1-\frac{(p^2)_\infty}{(p^2)_1}\right).$$

The paper presents this as the cubic cyclic specialization of the Cohen-Martinet heuristic, in agreement with Miller's preceding density conjecture. No resolution is given in the supplied text.