Kolmogorov superposition theorem states that a continuous function $f(x_1,\ldots,x_n)$ can be written as

$$f(x_1,\ldots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^{n}\phi_{q,p}(x_p)\right)$$

for certain continuous functions $\Phi_q$ and $\phi_{q,p}$.

For smooth, $f\in C^{\infty}$, can we obtain such a representation with the $\Phi_q$ and $\phi_{q,p}$ smooth too? Moreover, I am particularly interested in the case in which more than smooth, $f$ belongs to some Denjoy-Carleman quasianalytic class, but the question about smoothness would be a point to start.

I don't assume independence of $\Phi_q$ and $\phi_{q,p}$ of their parameters or of $f$.