Uniform lower-bound conjecture for the Schwarz preconditioner

Let $A$ be the discretized integral operator in the referenced two- or three-dimensional models, and let $T^{-1}_{\text{Schwarz}}$ be the Schwarz preconditioner associated with the specified overlapping domain decomposition. Uniform lower-bound conjecture. The smallest eigenvalue satisfies

$$\lambda_{\min}(T^{-1}_{\text{Schwarz}} A) \ge C > 0,$$

where $C$ is a positive constant. This conjecture asserts that the Schwarz preconditioner remains uniformly effective as the discretization is refined; the supplied text gives numerical motivation but no resolution.