  Extending earlier work by Sommers and Tymoczko, in 2016 Abe, Barakat, Cuntz,
Hoge, and Terao established that each arrangement of ideal type
$\mathcal{A}_\mathcal{I}$ stemming from an ideal $\mathcal{I}$ in the set of
positive roots of a reduced root system is free. Recently, R\"ohrle showed that
a large class of the $\mathcal{A}_\mathcal{I}$ satisfy the stronger property of
inductive freeness and conjectured that this property holds for all
$\mathcal{A}_\mathcal{I}$. In this article, we confirm this conjecture.
