Convex-hull conjecture for Orthomin(1)

Let $\zeta_1,\dots,\zeta_d\in\mathbb{C}$ be the spectral points used in the finite-dimensional Orthomin analysis, and let $\operatorname{Hull}(\zeta_1,\dots,\zeta_d)$ denote their convex hull. Let $q_n$ be the Orthomin$(1)$ residual convergence factor and let $\rho$ be the parameter in the associated linear system. Convex-hull conjecture. If

$$-\rho\in\operatorname{Hull}(\zeta_1,\dots,\zeta_d),$$

then

$$q_n\to\rho.$$

The paper describes this as a sharpened version of its earlier conjecture. The preceding proposition proves only the complementary obstruction that $-\rho$ outside the convex hull implies $q_n$ does not converge to $\rho$; the stated implication is therefore left open in the supplied text.