Let $A$ be a finite dimensional algebra with finite global dimension $g$ and $n$ simple modules. Let $d_1<d_2<...<d_r$ be the sequence of projective dimension of simple $A$-modules in increasing order. Define $\phi_A:= max \{ d_{i+1}-d_i | 1 \leq i \leq r-1 \}$.

Question: Is there a class of examples where $\phi_A$ gets arbitrary large for a fixed $n$?

I did not even find an example with $\phi_A >2$ (feel free to post an example if you know one).

Here are two examples for $\phi_A$:

In Projective dimensions of simple modules in acyclic quiver algebras , Jeremy Rickard proved that $\phi_A=1$ for acyclic quiver algebras.

Dag Madsen showed me an argument, which shows that $\phi_A \leq 2$ for any Nakayama algebra (this gives also a very nice proof that the global dimension of a Nakayama algebra with $n$ simple modules is at most $2n-2$).