Kashio–Yoshizaki refinement for traces of positive relative units

Let $\mathbb{B}_n$ be the maximal real subfield of $\mathbb{Q}(\zeta_{2^{n+2}})$. For the norm map ${\rm N}_{n/n-1}:E_n\to E_{n-1}$, set $RE_n^+=\ker({\rm N}_{n/n-1})$, and define ${\rm Tr}_n(\epsilon^2)=\sum_{\sigma\in{\rm Gal}(\mathbb{B}_n/\mathbb{Q})}\sigma(\epsilon^2)$. Let $c_1=2$ and, for $n\geq2$, let $c_n=2\cdot{\rm round}(2^n/5)$, where ${\rm round}(x)$ is the nearest integer to $x$. Kashio–Yoshizaki's refinement. For all $n\geq1$,

$$\min\{{\rm Tr}_n(\epsilon^2)\mid \epsilon\in RE_n^+\setminus \{\pm1\}\}=2^n(1+8c_n).$$

This is presented as a refinement of the Komatsu–Morisawa–Okazaki conjecture and gives an exact minimum for the positive relative units.