The canonical form conjecture for minimally dependent shifted elements

Let $\Lambda$ be a finitely generated subgroup of $\overline{\mathbb{Q}}^\times$ containing $\zeta_{m-1}$, with $m\ge 2$. Say that algebraic numbers are multiplicatively dependent modulo $\Lambda$ when their images satisfy a multiplicative relation modulo $\Lambda$. Let $x_1,\ldots,x_m\in\Lambda$ be distinct from $1$, and suppose that $x_1-1,\ldots,x_m-1$ are multiplicatively dependent modulo $\Lambda$, while every $m-1$ of them are multiplicatively independent modulo $\Lambda$. Canonical form conjecture. With finitely many exceptions, after a permutation,

$$x_k=\left(\zeta_{m-1}^{k-1}x_1\right)^{\pm1}\quad (k=2,\ldots,m-1),\qquad x_m=x_1^{\pm(m-1)}.$$

The claim gives a conjectural description of minimally dependent tuples in a finitely generated multiplicative group; the source does not provide a resolution.