  For every $n \geq 1$, we present examples of algebras $A$ having dominant
dimension $n$, such that the algebra $B=End_A(I_0 \oplus \Omega^{-n}(A))$ has
dominant dimension different from $n$, where $I_0$ is the injective hull of
$A$. This gives a counterexample to conjecture 2 of Chen and Xi. While the
conjecture is false in general, we show that a large class of algebras
containing higher Auslander algebras satisfies the property in the conjecture.
