Tail statistics of a sum of heavy-tailed random matrix ensembles

Let $X_1,\ldots,X_L$ be independently (not necessarily identically) distributed random matrices with heavy tails and unitary invariance, meaning that their eigenvalues and eigenvectors are uncorrelated. Assume that the largest eigenvalues occur on a scale larger than that of the bulk eigenvalues. Tail statistics of a sum of heavy-tailed random matrix ensembles. The statistics of the largest eigenvalues in the tail of the sum

$$\sum_{j=1}^L X_j$$

and of the direct sum

$$\bigoplus_{j=1}^L X_j$$

should be the same up to a scaling. The conjecture extends the observed tail-clustering behavior from identically distributed products of inverse Ginibre matrices to independently distributed heavy-tailed unitarily invariant ensembles; its validity for more general ensembles and for real or quaternionic matrices remains open.