Real Gaussian-divisible multi-eigenvector eigenvalue-repulsion bound

Prove the bound in Proposition for real Gaussian-divisible matrices when the statistic involves at least two eigenvectors, namely establish that, for real Gaussian-divisible matrices and m\geq2, the quantity L_{m,x,\mathbf{k},D,J}(W) satisfies the stated eigenvalue-repulsion estimate with the factor N^{-c\delta\sum_a k_a^2}.

Background

Proposition p-gauss-divisible establishes an upper bound for the joint contribution of configurations in which one or more localization intervals contain additional eigenvalues. The proposition is proved for real or complex Gaussian-divisible matrices when m=1, and for complex Gaussian-divisible matrices when m\geq2. The factor N^{-c\delta\sum_a k_a^2} reflects eigenvalue repulsion and is needed in the subsequent comparison argument for rare events involving several eigenvectors.

The unresolved case is the corresponding estimate for real Gaussian-divisible matrices with m\geq2. Establishing it would remove a technical restriction in the proof and would support the expected extension of the multi-eigenvector Gaussian-divisible analysis from complex to real matrices.