Iyama and Solberg introduced minimal Auslander-Gorenstein algebras as algebras having finite dominant dimension ($\geq 2$) equal to the Goreinstein dimension in https://www.sciencedirect.com/science/article/pii/S0001870816315055 . Such algebras with finite global dimension are exactly the higher Auslander algebras.
Questions:
In case a minimal Auslander-Gorenstein algebra has infinite global dimension, does it have Cartan determinant not equal to one?
In case a minimal Auslander-Gorenstein algebra has finite global dimension, does it have Cartan determinant equal to one?
I would think that the first question is false, but I was not able to find a counterexample yet.
