Find radius of convergence for a complicated series for $f'(f(x)) = f(f'(x))$

Question

Several months ago, I answered this question asking for solutions to the functional equation $f'(f(x)) = f(f'(x))$ by expanding as a formal Taylor series around some arbitrary fixed point of $f$. This gives a formal solution, and if the series is convergent it gives an analytic solution. I'm wondering if there's a way to prove that this series has nonzero radius of convergence? I tried some crude tricks (like inductively proving $|f^{(n)}(c)| < n! a^n$ for some $a$) but didn't get anywhere. Proving existence of an analytic solution would also suffice, but I haven't found a simpler way to prove that either, except in some special cases. Is there a way to prove non-zero radius of convergence for this series given arbitrary $c$ and $\lambda$?