Kummer's asymptotic conjecture for the relative class number

Let $q$ be a prime number, and let $h_1(q)$ denote the relative class number. Define

$$G(q):=2q\Bigl(\frac{q}{4\pi^2}\Bigr)^{\frac{q-1}{4}},\qquad R(q):=\frac{h_1(q)}{G(q)}.$$

The ratio $R(q)$ is called the Kummer ratio. Kummer's conjecture. As $q$ tends to infinity,

$$R(q)\longrightarrow 1.$$

Kummer made this conjecture in 1851; the paper notes that the ratio is close to $1$ for a generic prime $q$, while its extremal behavior remains the focus of study.