I'm working on a problem related to $\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof yet. I'm pretty sure this is a well known fact in hyperplane arrangements theory. All rigorous definitions and notations are available in Randell's original paper and in this expository notes by Stanley.
$\textbf{Question}$ Let $\mathcal{A}$ and $\mathcal{B}$ two complex hyperplanes arrangements in $\mathbb{C}^{d}.$ If the essentializations $\operatorname{ess}(\mathcal{A})$ and $\operatorname{ess}(\mathcal{B})$ are lattice-isotopic, then the arrangements $\mathcal{A}$ and $\mathcal{B}$ are lattice-isotopic.