Conjecture on the Kotake-Narasimhan theorem for ultradifferentiable structures

Let $\mathcal{U}$ be an ultradifferentiable structure that is closed under derivation and invariant under composition with mappings of class $\mathcal{U}$. The Kotake-Narasimhan theorem should hold in the class $\mathcal{U}$. The preceding results establish the theorem under more restrictive assumptions on the ultradifferentiable class, motivating the question of whether closure under derivation and invariance under composition are sufficient.