Characterization of essentially lexicographic valuations by separated completions

Let $\nu:F^*\rightarrow\mathbb{Z}^n$ be a maximal valuation, and let $\mathcal{O}=\{f\in\mathscr{F}:\nu(f)\geq 0\}$ with maximal ideal $\mathcal{m}=\{f\in\mathscr{F}:\nu(f)>0\}$ in the completion $\mathscr{F}$ of $F$ with respect to $\nu$. A total order on $\mathbb{Z}^n$ is essentially lexicographic if it belongs to the $GL(n,\mathbb{Z})$-orbit of the standard lexicographic order. Characterization conjecture. The valuation $\nu$ is associated with an essentially lexicographic order on $\mathbb{Z}^n$ if and only if

$$\bigcap_{i>1}\mathcal{m}^i=0.$$

This condition characterizes when the powers of the maximal ideal have zero intersection, linking the order structure of the value group to the separatedness of the associated valuation ring. The supplied text does not state whether the assertion has been proved or disproved.