Let $A$ be a finite dimensional symmetric algebra (given by a connected quiver and non-semisimple) and $S$ a simple $A$-module. (I am more interested in symmetric algebras, but selfinjective examples are also welcome)

Questions:

• Is there an easy example with $Ext_A^i(S,S)=0$ for all $i \geq m$ for some $m$? (easy might mean that the example also works over finite fields)

• Can we have $Ext_A^m(S,S) \neq 0$ for exactly one $m$? (In case $m=1$ this would give a counterexample to the extreme no loop conjecture)