Let $A$ be a finite dimensional algebra with finite global dimension $g$ and $e$ the idempotent such that $eA$ is the direct sum of all indecomposable projective-injective $A$-modules. Do we have $g=injdim(A/AeA)?$

I am mostly interested in the case when $eA$ is injective and faithful, which is also the only case where I really tested it. I tested it for Nakayama algebras and found no counterexamples. I found a proof for higher Auslander algebras. Note that this is also true in case there are no projective-injective modules since then $A/AeA=A$ and the assertion is then well-known. It is not true for algebras of infinite global dimension: The Nakayama algebra with Kupisch series [3,4] has infinite global dimension but $A/AeA$ has injective dimension equal to one.