The determinant- $p$ conjecture for twisted derivations of cyclotomic integer rings

Let $p$ be an odd prime, let $K=\mathbb{Q}(\zeta)$ be the $p$th cyclotomic field, and write $O_K=\mathbb{Z}[\zeta]$. For $\beta=\sum_{i=0}^{p-2}b_i\zeta^i\in O_K$, suppose

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

The determinant-$p$ conjecture. The matrix $A=[a_{ij}]$ is a $(p-1)\times(p-1)$ matrix with determinant $p$.

The conjecture has been verified computationally for all odd primes below $100$, but remains unproved for general odd primes.