Let an algebra always be finite dimensional over a field and connected. It is well known that a quasi-hereditary algebra with $n$ simple modules has global dimension at most $2n-2$.
Questions: 1. Do algebras derived equivalent to quasi-hereditary algebras with $n$ simple modules also have global dimension at most $2n-2$?
Is there a good discrete test (for example with a computer) for checking whether an algebra of finite global dimension is derived equivalent to a quasi-hereditary algebra?
Is every Nakayama algebra of finite global dimension derived equivalent to a quasi-hereditary algebra?
For example the Nakayama algebras with $n$ simples and Kupisch series $[n,n+1,...,n+1]$ are not quasi-hereditary for $n \geq 3$, but they are derived equivalent to the representation-finite block of a Schur algebra, which is quasi-hereditary. Here are some Kupisch series of Nakayama algebras of finite global dimension that are not quasi-hereditary, but Im not sure whether they are derived equivalent to a quasi-hereditary algebra: [ [ 3, 3, 3, 4 ], [ 3, 5, 5, 4 ], [ 4, 5, 6, 5 ], [ 4, 6, 5, 5 ], [ 4, 6, 6, 5 ] ]
It would be interesting to see how to deal with this problem even in small examples.