Outer-derivation criterion for conductor $p^k$

Let $\mathbb{K}=\mathbb{Q}(\zeta)$ be the $p^k$th cyclotomic number field, where $k\in\mathbb{N}\setminus\{1\}$ and $p$ is a rational prime, and let $O_{\mathbb{K}}=\mathbb{Z}[\zeta]$. Let $u,v\in U(p^k)$ 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

$$D(\zeta)=\sum_{i=0}^{p^{k-1}(p-1)-1}c_i\zeta^i.$$

Outer-derivation conjecture. Under the hypotheses of the determinant conjecture, $D$ is outer if and only if

$$\frac{1}{p^{e_1}}\operatorname{Adj}(A)C\notin\mathbb{Z}^{p^{k-1}(p-1)}.$$

In particular, non-zero outer derivations exist, equivalently $\operatorname{Inn}_{(\sigma,\tau)}(O_{\mathbb{K}})\subsetneq\mathcal{D}_{(\sigma,\tau)}(O_{\mathbb{K}})$. This conjecture proposes the outer-derivation counterpart to the innerness criterion for prime-power conductor.