The Weber class-number conjecture for the fields $\mathbb{B}_n$

Let $\mathbb{B}_n$ be the number field introduced above, and let $h_n$ denote its class number for $n\in\mathbb{Z}_{\geq 1}$. Weber's conjecture. For every $n\in\mathbb{Z}_{\geq 1}$, one has

$$h_n=1.$$

This is the class-number-one assertion commonly associated with the Weber conjecture; the supplied source does not establish its resolution.