The class-number distribution conjecture for prime-generated real quadratic fields

Let $p_i$) be the $i$-th prime, let $\mathfrak{P}=\{\mathbb{Q}(\sqrt{p})\mid p\text{ prime}\}$, and define

$$\mathbb{P}(h=q)=\lim_{x\to\infty}\frac{\#\{i\mid h(\mathbb{Q}(\sqrt{p_i}))=q,\ i\leqslant x\}}{x}.$$

Assume these probabilities exist for all positive odd integers $q$. Write $\mathbb{P}(h=1)=\mathbb{P}_0$ and $\mathbb{P}(h=q)=\lambda_q\mathbb{P}_0$, so that $\lambda_1=1$. The class-number distribution conjecture. For every odd prime $p$,

$$\lambda_p=\frac{1}{p(p-1)},\qquad \lambda_{p^n}=\lambda_{p^{n-1}}\left(p^{\lfloor n/2+1\rfloor}-1\right)^{-1}\quad(n\geqslant2),$$

and if $q=p_1^{r_1}\cdots p_t^{r_t}$, then

$$\lambda_q=\lambda_{p_1^{r_1}}\cdots\lambda_{p_t^{r_t}}.$$

The conjecture predicts the relative frequencies of every positive odd class number in this family. The paper states that it implies that more than $75\%$ of these fields have class number one, and hence implies Gauss's real quadratic class-number conjecture; it also notes that the claim is far from proved and that numerical convergence is slow.