Is there a bound on the global dimension for higher Auslander algebras with n simple modules? I think I am not aware of an example where the global dimension is larger than 2n-2.

Let algebras be finite dimensional and connected. Recall that an algebra $A$ is called a higher Auslander algebra in case it the dominant dimension coincides with the global dimension and both dimension are finite and larger than or equal to two.

Those algebras were introduced by Iyama as a generalisation of the classical Auslander algebras in https://www.sciencedirect.com/science/article/pii/S0001870806001733 .

Question: Is there a bound on the global dimension for higher Auslander algebras with n simple modules? I think I am not aware of an example where the global dimension is larger than 2n-2.