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Star moments convergence of random matrices

Let $latex (X_{jk})_{j,k\geq1}$ be an infinite table of i.i.d. complex random variables with positive variance and all moments bounded by a constant. Set $latex M:=(X_{jk})_{1\leq j,k\leq n}$. I do believe that with probability one and in expectation, the $latex *$-moments of $latex M$ converge as $latex n\to\infty$ to the $latex *$-moments of the Voiculescu circular element. This result is well known when $latex X_{11}$ is Gaussian. There is maybe a proof written somewhere, involving some paths combinatorics. Can you link the question with the work of Nourdin and Peccati?

If it holds, this statement shows that the hypothesis in the Śniady theorem is always satisfied by these random matrices. Moreover, the regularization by a Ginibre Ensemble is not needed thanks to the Tao and Vu bound on the smallest singular values. Well, TT will probably say a word on this in his recent course.

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