Uniform condition-number conjecture for the trace Schur complement

Let $\lambda_{\min}(s,h)$ and $\lambda_{\max}(s,h)$ be respectively the smallest and largest generalized eigenvalues of the trace Schur-complement matrix relative to the fractional norm matrix ${\mathsf{H}}_s$, with $s<0$. Define the condition number by

$$\kappa(s,h)=\frac{\lambda_{\max}(s,h)}{\lambda_{\min}(s,h)}.$$

Uniform condition-number conjecture. There exist $s<0$ and a constant $C$ such that

$$\kappa(s,h)\leq C\qquad\forall h>0.$$

This weaker requirement asks only for a mesh-independent condition number, rather than uniform lower and upper spectral-equivalence bounds; the source presents it as an additional computational criterion.