Conjecture that finite-dimensional algebras have property *

Let $A$ be a finite dimensional algebra with finite dominant dimension $n\geq 1$. Say that $A$ has property $*$ if, for a minimal injective resolution

$$0\rightarrow A\rightarrow I_0\rightarrow I_1\rightarrow \dots,$$

the algebra

$$B:=\operatorname{End}_A(I_0\oplus \Omega^{-n}(A))$$

has dominant dimension $n$.

Dominant-dimension property conjecture. Every finite dimensional algebra with finite dominant dimension $n\geq 1$ has property $*$.

The conjecture was disproved in this paper by examples showing that the associated algebra $B$ can have dominant dimension different from $n$. The property nevertheless holds for important subclasses, including Morita algebras and a large class containing higher Auslander algebras.