Innerness criterion for twisted derivations when the conductor is $2^r p$

Let $\mathbb{K}=\mathbb{Q}(\zeta)$ be the $n$th cyclotomic number field with $n=2^r p$, where $r\in\mathbb{N}$ and $p$ is an odd rational prime, and let $O_{\mathbb{K}}=\mathbb{Z}[\zeta]$. Let $u,v\in U(n)$ with $u\ne v$, let $\sigma(\zeta)=\zeta^u$ and $\tau(\zeta)=\zeta^v$, and let $A$ be the coefficient matrix from the preceding determinant conjecture. For a $(\sigma,\tau)$-derivation $D:O_{\mathbb{K}}\to O_{\mathbb{K}}$, write

$$D(\zeta)=\sum_{i=0}^{2^{r-1}(p-1)-1}c_i\zeta^i\in O_{\mathbb{K}},\qquad C^T=(c_0,\ldots,c_{2^{r-1}(p-1)-1}).$$

Innerness conjecture. Under the hypotheses of the determinant conjecture, if $1\le e_1\le r-1$ and $e_2\ge1$, then $D$ is inner if and only if

$$\frac{1}{2^{2^{e_1}(p-1)}}\operatorname{Adj}(A)C\in\mathbb{Z}^{2^r(p-1)}.$$

In particular, divisibility of every $c_i$ by $2^{2^{e_1}(p-1)}$, and even divisibility of every $c_i$ by $2$, implies innerness. If $e_1\ge r$ and $e_2=0$, replace the denominator by $p^{2^{r-1}}$; in all other cases $D$ is always inner. This is the proposed necessary-and-sufficient characterization of inner twisted derivations for this conductor form; together with the determinant claim, it is intended to distinguish the cases where outer derivations can occur.