Let $A$ be a finite dimensional algebra. For a subcategory $C$ of $mod-A$ let $\overline{C}$ be the objects $X \in mod-A$ such that there exists an exact sequence $0 \rightarrow C_n \rightarrow ... \rightarrow C_0 \rightarrow X \rightarrow 0$ and let $\underline{C}$ be defined dually. In the article "Tilting theory and homologically finite subcategories with applications to quasihereditary algebras" by Idun Reiten, it is mentioned that for a tilting module $T$ it is not known whether the functor $D Hom_A(-,T) : \underline{addT} \rightarrow \overline{add(D(T))}$ is an equivalence in general. It is noted that it is an equivalence in case $A$ has finite global dimension.

Question : Is there progress on this open problem since the article was released? In particular, is it known whether this is an equivalence in some other cases like when $A$ has finite finitistic dimension or is representation-finite?

Let $A$ be a finite dimensional algebra. For a subcategory $C$ of $mod-A$ let $\overline{C}$ be the objects $X \in mod-A$ such that there exists an exact sequence $0 \rightarrow C_n \rightarrow ... \rightarrow C_0 \rightarrow X \rightarrow 0$ and let $\underline{C}$ be defined dually. In the article "Tilting theory and homologically finite subcategories with applications to quasihereditary algebras" by Idun Reiten, it is mentioned that for a tilting module $T$ it is not known whether the functor $D Hom_A(-,T) : \underline{addT} \rightarrow \overline{add(D(T))}$ is an equivalence in general. Note that for a tilting module $T$ we have $\underline{add(T)}=^{\perp}(T^{\perp})$ and for the $B$-cotilting module $D(T)$ we have $\overline{add(D(T))}=(^{\perp}(D(T)))^{\perp}$, when $B=End_A(T)$.

It is noted that it is an equivalence in case $A$ has finite global dimension.

Question : Is there progress on this open problem since the article was released? In particular, is it known whether this is an equivalence in some other cases like when $A$ has finite finitistic dimension or is representation-finite?