Non-lifting idempotents in the quotient category overline{h𝒯_∞}(T^*M)

Determine whether there exists an idempotent in the quotient category \overline{h𝒯_∞}(T^*M) (obtained by modding out morphisms of infinite spectral value) that does not lift to an idempotent in h𝒯_∞(T^*M); equivalently, construct such an example or prove that every idempotent in \overline{h𝒯_∞}(T^*M) lifts.

Background

To refine statements about spectral invariants, the paper introduces a 1-category \overline{h𝒯_∞}(T^*M) by quotienting morphisms whose spectral value c(−; F, G) is +∞. In reasonable cases this quotient coincides with the original homotopy category, but in general it need not.

While idempotents in h𝒯∞(T^*M) induce idempotents in the quotient, it is unknown whether there exist genuinely new idempotents in the quotient that do not come from h𝒯∞(T^*M). Resolving this would clarify whether the quotient introduces additional splitting phenomena not present in the original category.