Brown representability for covariant functors on abstract well generated triangulated categories

Determine whether Brown representability holds for covariant additive functors on arbitrary well generated triangulated categories; namely, ascertain if an additive functor F: T → Ab that is homological and commutes with products is corepresentable when T is a well generated triangulated category without assuming T ≅ hC for a presentable stable ∞-category C.

Background

Section 1.3 recalls that presentable stable ∞-categories correspond to well generated triangulated categories and reviews Brown representability results. For contravariant functors, Brown representability holds in well generated settings. For covariant functors, the paper notes known positive results in the case when T ≅ hC with C presentable stable, using Neeman’s theorems and Lurie’s framework.

However, the authors explicitly state that, beyond the presentable stable setting, it is unknown whether Brown representability for covariant functors holds in abstract well generated triangulated categories, highlighting a longstanding gap.