Tilting Completion and Self-Orthogonality Modules

We give negative answers to two tilting-completion questions for finite-dimensional algebras. We construct two finite-dimensional basic connected quasi-hereditary $\mathbb C$-algebras. The first admits a faithful basic full-rank pretilting module with no tilting completion; the second admits an almost-tilting module with no tilting completion. The full-rank example also yields counterexamples to two conjectures: Enomoto's Self-orthogonal Wakamatsu-tilting Conjecture and the Self-orthogonal Faithful Conjecture of Chen, Li, Zhang, and Zhao. We further show that the Self-orthogonal Wakamatsu-tilting Conjecture holds for all finite-dimensional algebras if and only if the Self-orthogonal Faithful Conjecture holds for all finite-dimensional algebras. The construction ultimately stems from Krah's non-full exceptional collection of maximal length on a rational surface and Kalck's associated full-rank presilting example.