Conformal Killing $2$-Forms on Compact Symmetric Spaces

We show that a simply connected compact symmetric space of dimension $n\ge 3$ admits a conformal Killing $2$-form which is not Killing if and only if it is homothetic to one of ${\mathbb{S}}^n$, ${\mathbb{C}\mathbb{P}}^m$ or ${\mathbb{H}\mathbb{P}}^q$.