Divisionally free ideal-subarrangements and ideal-Shi arrangements

Let $\Phi^+$ be the set of positive roots of a root system, with $\alpha\geq\beta$ when $\alpha-\beta$ is a nonnegative linear combination of simple roots. An ideal is a subset $I\subset\Phi^+$ such that $\alpha\in I$ and $\alpha\geq\beta$ imply $\beta\in I$. Define the ideal-subarrangement and ideal-Shi arrangements by

$${\mathcal{A}}_I=\{H_\alpha^0\mid\alpha\in I\},$$ $$\operatorname{Shi}^k_{+I}=\operatorname{Shi}^k\cup\{H_\alpha^{-k}\}_{\alpha\in I},\qquad \operatorname{Shi}^k_{-I}=\operatorname{Shi}^k\setminus\{H_\alpha^{k}\}_{\alpha\in I}.$$

The ideal-arrangement conjecture. All ideal-subarrangements ${\mathcal{A}}_I$ and ideal-Shi arrangements $\operatorname{Shi}^k_{\pm I}$ are divisionally free.

This would unify the divisional freeness of the ideal-subarrangements and ideal-Shi arrangements considered in the cited work. The source gives the claim as a conjecture based on known families, and no resolution is supplied here.