The asymptotic conjecture for the normalized sum involving $J_n$

Let $n$ range over positive integers such that $p=4n-1$ is prime, and let $J_n$ be the quantity defined earlier in the paper. Asymptotic conjecture for $J_n$.

$$\lim_{n\rightarrow\infty}\frac{J_n}{n}=\frac{3}{8}.$$

The paper presents this as a conjecture based on numerical data concerning class-number identities for the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$. No resolution is supplied in the given text.