A free analogue of Shannon's problem on monotonicity of entropy

We prove a free probability analog of a result of [S. Artstein, K. Ball, F. Barthe, A. Naor, Solution of Shannon's problem on monotonicity of entropy, J. Amer. Math. Soc. 17 (2004) 975-982]. In particular, we prove that if X1,X2,… are freely independent identically distributed random variables, then the functionn↦χ(X1+⋯+Xnn) is monotone increasing for all n.