A parity obstruction to completeness of object cotorsion pairs

We construct a complete ideal cotorsion pair of object ideals in a Hom-finite, weakly idempotent complete Frobenius exact category whose associated cotorsion pair of objects is neither special precovering nor special preenveloping. The category consists of bounded complexes of finite-dimensional vector spaces with even total cohomology dimension. Special ideal approximations are given by explicit cone sequences, whereas special object approximations are obstructed by the parity of truncated cohomology. This gives a negative answer to Question 29 of Fu, Guil Asensio, Herzog and Torrecillas even under weak idempotent completeness and a Frobenius hypothesis.