Komatsu–Morisawa–Okazaki trace conjecture for relative units

Let $\mathbb{B}_n$ be the maximal real subfield of $\mathbb{Q}(\zeta_{2^{n+2}})$, let $E_n$ be its group of units, and let ${\rm N}_{n/n-1}:E_n\to E_{n-1}$ be the norm map. Define

$$RE_n={\rm N}_{n/n-1}^{-1}({\{\pm1\}}),$$

and, for $\epsilon otin\{\pm1\}$, define ${\rm Tr}_n(\epsilon^2)=\sum_{\sigma\in{\rm Gal}(\mathbb{B}_n/\mathbb{Q})}\sigma(\epsilon^2)$. Komatsu–Morisawa–Okazaki's conjecture. For all $n\geq1$,

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

Morisawa and Okazaki resolved the assertion for $RE_n^-$, but the corresponding assertion for $RE_n^+$ remains unresolved. This conjecture concerns traces of relative units in the cyclotomic tower and is relevant to Weber's class number problem.