Spectral-gap-independent constants for randomized block polynomial interpolation

Fix integers $b \ge 1$ and $d \ge 2$, and call diagonal matrices $\Lambda_1, \dots, \Lambda_d \in \mathbb{R}^{b \times b}$ with pairwise disjoint spectra admissible. Draw all entries of $\Omega_1, \dots, \Omega_d \in \mathbb{R}^{b \times b}$ independently from $N(0, 1)$, put $B_i = \Omega_i^{-1}\Lambda_i\Omega_i$, and let $a$ and $c$ be the smallest and largest diagonal entries of all the $\Lambda_i$. For each $k \in \{1, \dots, d\}$, relabel the triples $(B_i, \Lambda_i, \Omega_i)$ in the order $(k, 1, \dots, k-1, k+1, \dots, d)$, mark the relabelled matrices with a superscript $(k)$, and compute, for $i = d, d-1, \dots, 1$ in turn, $$S^{(k)}_{i,i} = I_b, \qquad S^{(k)}_{i,j} = B^{(k)}_i S^{(k)}_{i,j-1} - S^{(k)}_{i,j-1}\widehat B^{(k)}_j \quad (j = i+1, \dots, d),$$ $$\widehat\Omega^{(k)}_i = \Omega^{(k)}_i S^{(k)}_{i,d}, \qquad \widehat B^{(k)}_i = \bigl(\widehat\Omega^{(k)}_i\bigr)^{-1} \Lambda^{(k)}_i \widehat\Omega^{(k)}_i.$$ With spectral norms, let $$\chi^{(k)}_{\mathrm{mono}} = \max_{2 \le i \le d} \left\{ 1, \frac{\|aI - \widehat B^{(k)}_i\|_2}{\|aI - \Lambda^{(k)}_i\|_2}, \frac{\|cI - \widehat B^{(k)}_i\|_2}{\|cI - \Lambda^{(k)}_i\|_2} \right\},$$ where a ratio $0/0$ counts as $1$, $$\chi^{(k)}_{\mathrm{coef}} = \bigl\|(S^{(k)}_{1,d})^{-1}\bigr\|_2^{1/(d-1)} \min_{\substack{2 \le i \le d,\ \lambda \in \sigma(\Lambda^{(k)}_1) \\ \eta \in \sigma(\Lambda^{(k)}_i)}} |\lambda - \eta|,$$ $\chi_{\mathrm{mono}} = \max_k \chi^{(k)}_{\mathrm{mono}}$ and $\chi_{\mathrm{coef}} = \max_k \chi^{(k)}_{\mathrm{coef}}$; Shao showed that the inverses involved exist almost surely.

Shao introduced these quantities in an analysis of randomized small-block Lanczos: their product enters his bound on the fundamental matrix polynomials of block polynomial interpolation outside an interval containing the spectra. When $b = 1$, or when every $\Lambda_i$ is a multiple of the identity, $\chi_{\mathrm{mono}} = 1$ and $\chi_{\mathrm{coef}} \le 1$. Shao conjectured, with numerical evidence, that with high probability $\chi_{\mathrm{mono}}\chi_{\mathrm{coef}}$ is bounded by a constant depending on $b$ and $d$ but not on the $\Lambda_i$, nor on the eigenvalue gaps within and between them.

Problem. Is $\chi_{\mathrm{mono}}\chi_{\mathrm{coef}}$ uniformly bounded in probability over all admissible data, that is, for each $0 < \delta < 1$, is there a finite $C(b, d, \delta)$ such that $$\Pr\{\chi_{\mathrm{mono}}\chi_{\mathrm{coef}} \le C(b, d, \delta)\} \ge 1 - \delta$$ for every fixed admissible $(\Lambda_1, \dots, \Lambda_d)$? The constant must be independent of the eigenvalues and of all gaps within and between the blocks, and the probability is taken separately for each fixed input.