Determinant conjecture for twisted derivations of 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 write

$$\beta=\sum_{i=0}^{p^{k-1}(p-1)-1}b_i\zeta^i\in O_{\mathbb{K}},\qquad \beta(\tau-\sigma)(\zeta)=\sum_{i=0}^{p^{k-1}(p-1)-1}\left(\sum_{j=0}^{p^{k-1}(p-1)-1}a_{ij}b_j\right)\zeta^i.$$

Let $A=[a_{ij}]$ and suppose $v-u=p^{e_1}m$, where $m$ is positive, $e_1$ is non-negative, and $p$ does not divide $m$. Determinant conjecture. The matrix $A$ is a $p^{k-1}(p-1)\times p^{k-1}(p-1)$ matrix with determinant $\pm p^{p^{e_1}}$. This conjectured determinant provides the arithmetic input for the proposed innerness criterion in the prime-power conductor case; the sign depends on the ordering of the rows.