Cluster robustness conjecture for randomized small-block Lanczos

Let $\Lambda_{1},\dotsc,\Lambda_{d}\in\mathbb{R}^{b\times b}$ be real diagonal matrices, and assume that $\Lambda_i$ and $\Lambda_j$ have disjoint spectra for $1\leq i<j\leq d$. Let $[\Omega_1,\dotsc,\Omega_d]$ be a Gaussian random matrix with $\Omega_i\in\mathbb{R}^{b\times b}$. The quantities $\chi_{\mathsf{mono}}$ and $\chi_{\mathsf{coef}}$, defined in the paper, measure the relevant polynomial-growth and coefficient effects in the cluster-robustness bound. Cluster robustness conjecture. With high probability, the product $\chi_{\mathsf{mono}}\chi_{\mathsf{coef}}$ is bounded by a constant depending only on $b$ and $d$, independently of the matrices $\Lambda_i$ for $1\leq i\leq d$. This would make the spectral-gap dependence in the cluster-robustness estimate uniform over the diagonal blocks, beyond the currently established upper bound.