Yoshizaki's generation conjecture for positive relative units

Let $\mathbb{B}_n$ be the maximal real subfield of $\mathbb{Q}(\zeta_{2^{n+2}})$, let $X_n=2\cos(2\pi/2^{n+2})$, and let $RE_n^+=\ker({\rm N}_{n/n-1}:E_n\to E_{n-1})$. Define

$$\epsilon_n=\frac{X_n+1}{X_n-1}.$$

The notation $\langle-1,\epsilon_n\rangle_{{\rm Gal}(\mathbb{B}_n/\mathbb{Q})}$ denotes the subgroup generated by $-1$ and $\epsilon_n$ as a ${\rm Gal}(\mathbb{B}_n/\mathbb{Q})$-module. Yoshizaki's generation conjecture. For all $n\geq1$,

$$RE_n^+=\langle-1,\epsilon_n\rangle_{{\rm Gal}(\mathbb{B}_n/\mathbb{Q})}.$$

This asserts that the explicitly constructed unit $\epsilon_n$ generates all positive relative units, together with $-1$, under the Galois action. The supplied text does not state whether the assertion has been resolved.