Nonexistence of cyclotomic norm representations for two candidate multipliers

Let $n$ be an even integer with $n\geq 16$ and $n\equiv 0\pmod{4}$. Let $\zeta_{2n}$ be a primitive $2n$th root of unity, let $\overline{\eta}$ denote complex conjugation, and let $\lambda_1$ be the multiplier under consideration. Nonexistence conjecture. There is no $\eta\in\mathbb{Z}[\zeta_{2n}]$ such that

$$\lambda_1=\eta\overline{\eta}=4+\zeta_{2n}+\overline{\zeta_{2n}}$$

or such that

$$\lambda_1=\eta\overline{\eta}=5+\zeta_{2n}+\overline{\zeta_{2n}}.$$

The statement concerns the possible cyclotomic inflation multipliers obtained as norms of elements of $\mathbb{Z}[\zeta_{2n}]$; the supplied text gives no resolution beyond the asserted nonexistence, so its status should be checked against the source.