Multiplicative independence conjecture for shifted roots of unity

Let $k$ be an odd prime, and let $\zeta_k$ denote a primitive $k$-th root of unity. For positive integers $m,n$ with $m>n$, consider the algebraic integers $-m+\zeta_k$ and $-n+\zeta_k$ in $\mathbb{Z}[\zeta_k]$.

Multiplicative independence conjecture. The algebraic integers

$$-m+\zeta_k\quad\text{and}\quad -n+\zeta_k$$

are multiplicatively independent.

This conjecture is the odd-prime case of the paper's goal of extending known multiplicative-independence results for bases of cyclotomic integer rings. The supplied source does not state whether this conjecture has been resolved.