We show that the empirical eigenvalue measure for sum of $d$ independent Haar distributed $n$-dimensional unitary matrices, converge for $n \rightarrow \infty$ to the Brown measure of the free sum of $d$ Haar unitary operators. The same applies for independent Haar distributed $n$-dimensional orthogonal matrices. As a byproduct of our approach, we relax the requirement of uniformly bounded imaginary part of Stieltjes transform of $T_n$ that is made in [Guionnet, Krishnapur, Zeitouni; Theorem 1].
"Limiting spectral distribution of sum of unitary and orthogonal matrices." Electron. Commun. Probab. 18 1 - 19, 2013. https://doi.org/10.1214/ECP.v18-2466