Equality of ghost index and level in arbitrary triangulated categories

Determine whether, for every triangulated category T with arbitrary direct sums and every full subcategory G of T that is closed under suspensions, the ghost index gin^G_T(X) equals the level level^G_T(X) for all objects X in T.

Background

The paper studies levels in triangulated categories and relates them to ghost maps via the ghost lemma. In general, one has the inequality gin^G_T(X) ≤ level^G_T(X). Under the additional hypothesis that G is contravariantly finite, equality holds; this is referred to as the converse ghost lemma.

The authors prove openness results by leveraging a version of the converse ghost lemma for compact objects (Lemma 2.14), but they note that the general question of whether ghost index and level coincide in arbitrary triangulated categories remains unresolved.