Poisson tail-statistics conjecture for heavy-tailed stable random matrices

Let $X$ be a heavy-tailed stable random matrix with unitary invariance, meaning that its eigenvalues and eigenvectors are uncorrelated. Suppose that the largest eigenvalues are considerably larger than the bulk eigenvalues. Poisson tail-statistics conjecture. The local spectral statistics in the tail of $X$ should follow Poisson statistics. The conjecture is motivated by numerical and analytical observations for averaged Gaussian unitary ensembles: Poisson statistics are expected when the largest-eigenvalue scale is much larger than the bulk scale, whereas otherwise a mixture involving sine-kernel statistics may occur. Its general validity remains open.