Conjecture on dominant dimensions of endomorphism algebras

Let $A$ be a finite dimensional algebra with finite dominant dimension $n\geq 1$. Let

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

be a minimal injective resolution of $A$, and set

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

Dominant-dimension conjecture. The algebra $B$ has dominant dimension $n$.

The claim concerns how dominant dimension behaves under this endomorphism-algebra construction. It is false in general: for every $n\geq 1$, there are counterexamples, although the property still holds for a large class of algebras including higher Auslander algebras.