Cohen–Lenstra–Soundararajan conjectural asymptotic for odd imaginary quadratic class numbers

For an odd positive integer $h$, let $\mathcal{F}(h)$ be the number of negative fundamental discriminants $d$ with class number $h(d)=h$. Let $\mathbb{Y}(p)$ be independent random variables taking the values $1$ and $-1$ with probability $1/2$, and set

$$L(1,\mathbb{Y})=\prod_p\left(1-\frac{\mathbb{Y}(p)}{p}\right)^{-1}.$$

Define

$$\mathfrak{C}=15\prod_{\substack{\ell\geq 3\\ \ell\text{ prime}}}\prod_{i=2}^{\infty}\left(1-\frac{1}{\ell^i}\right)$$

and, for odd $h$,

$$\mathfrak{c}(h)=\prod_{p^n\parallel h}\prod_{i=1}^n\left(1-\frac{1}{p^i}\right)^{-1}.$$

The refined Soundararajan conjecture. As $h\to\infty$ through odd values,

$$\mathcal{F}(h)\sim\frac{\mathfrak{C}}{15}\mathfrak{c}(h)h\,\mathbb{E}\left(\frac{1}{L(1,\mathbb{Y})^2\log(\pi h/L(1,\mathbb{Y}))}\right)\sim\mathfrak{C}\mathfrak{c}(h)\frac{h}{\log(\pi h)}.$$

This refines Soundararajan's heuristic $\mathcal{F}(h)\asymp h/\log h$ and combines Cohen–Lenstra heuristics with the distribution of quadratic Dirichlet $L$-values. The source presents it as conjectural; its resolution status is not specified here.