A new Berry-Esseen-type estimate in the free central limit theorem

Using the subordination approach, we provide a new Berry-Esseen-type estimate in the free central limit theorem in terms of the fourth Lyapunov fraction. In the special case of identical distributions, our result implies a rate of order $n^{-1/2 + \varepsilon}$ for any $\varepsilon>0$, thus almost leading to the optimal rate of order $n^{-1/2}$.