Cohen-Lenstra-Martinet density heuristics for elementary abelian 2- and 3-extensions

Let $t$ be a positive integer, let $p$ be an odd prime in the degree-$2$ case, and let $p\geq 5$ be prime in the degree-$3$ case. Write $(p)_\infty=\prod_{k\geq1}(1-p^{-k})$ and $(p)_1=1-p^{-1}$, and analogously $(p^2)_\infty=\prod_{k\geq1}(1-p^{-2k})$ and $(p^2)_1=1-p^{-2}$. Cohen-Lenstra-Martinet heuristic. If $K=\mathbb{Q}(\sqrt{d_1},\ldots,\sqrt{d_t})$, then

$$\mathrm{Prob}(p\nmid h_K)=\left(\frac{(p)_\infty}{(p)_1}\right)^{2^t-1}.$$

If $K$ has degree $3^t$ and is the compositum of $t$ cyclic cubic fields, then

$$\mathrm{Prob}(p\nmid h_K)= \begin{cases} \left(\dfrac{(p)_\infty}{(p)_1}\right)^{2(3^t-1)/2},&p\equiv1\pmod3,\\ \left(\dfrac{(p^2)_\infty}{(p^2)_1}\right)^{(3^t-1)/2},&p\equiv2\pmod3. \end{cases}$$

These are heuristic predictions obtained by treating the class groups of intermediate cyclic prime-degree fields as independent. The source does not state a resolution.