The adjugate criterion for inner twisted derivations

Let $p$ be an odd prime, let $K=\mathbb{Q}(\zeta)$ with $O_K=\mathbb{Z}[\zeta]$, and let $A$ be the matrix defined by

$$\beta(\tau-\sigma)(\zeta)=\sum_{i=0}^{p-2}\left(\sum_{j=0}^{p-2}a_{ij}b_j\right)\zeta^i$$

for $\beta=\sum_{i=0}^{p-2}b_i\zeta^i\in K$. Let $D$ be a $(\sigma,\tau)$-derivation of $O_K$ with

$$D(\zeta)=\sum_{i=0}^{p-2}c_i\zeta^i\in O_K,$$

and write $C^T=(c_0\ c_1\ \ldots\ c_{p-2})$. The adjugate criterion. With these notations, $D$ is inner if and only if

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

In particular, $O_K$ has non-trivial outer $(\sigma,\tau)$-derivations, and if $p$ divides every $c_i$, then $D$ is inner. The source presents this as a consequence of the determinant conjecture and reports no proof of the conjecture itself.