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A recent work by A. Mandarino, T. Linowski and K. \.{Z}yczkowski left openthe following question. If $ \mu_N $ is a certain permutation of entries of a $N^2 \times N^2 $ matrix ("mixing map") and $ U_N $ is a $ N^2 \times N^2 $ Haarunitary random matrix, then is the family $ U_N, U_N^{\mu_N}, ( U_N^2)^{\mu_N}, \dots , ( U_N^m)^{\mu_N} $ asymptotically free? (here by $A^{ \mu}$we understand the matrix resulted by permuting the entries of $ A $ accordingto the permutation $ \mu $). This paper presents some techniques forapproaching such problems. In particular, one easy consequence of the mainresult is that the question above has an affirmative answer.

Answer to a question by A. Mandarino, T. Linowski and K. \.{Z}yczkowski