Failure of spectrum-uniform probabilistic interpolation bounds

For block size $b = 2$ and $d = 3$ blocks, the manuscript gives deterministic
diagonal data with pairwise disjoint spectra, $\Lambda_1 = 2I_2$,
$\Lambda_2 = \operatorname{diag}(\varepsilon, 2\varepsilon)$ and
$\Lambda_3 = \operatorname{diag}(0, 1)$, for which the random interpolation
constant $\chi_{\mathrm{mono}}\chi_{\mathrm{coef}}$ from Shao's analysis of
randomized small-block Lanczos is not bounded in probability uniformly in
$\varepsilon$: along $\varepsilon_m = 1/(m+5)$, the probability that it is at
most any fixed finite $C$ tends to zero. Hence no spectrum-independent
constant exists, already for failure probability $1/2$. The proof follows the
literal recurrence. A $2 \times 2$ determinant expansion shows that the
transformed second block tends almost surely to a nonzero nilpotent matrix as
$\varepsilon \to 0$, so the monomial factor is at least
$\|\widehat B_2\|_2/(2\varepsilon) \to \infty$, while the coefficient factor
stays at least $8^{-1/4}$. A polynomial null-set argument shows that every
prescribed ordering is defined almost surely.