Let $A$ be a (connected) finite dimensional algebra with Jacobson radical $J$.

>Question: Is the sequence $id(J^i)$ for $i=1,2,...,$ monotone decreasing? 

(one can ask the same question for $pd(J^i)$.) 

Surprisingly, computer experiments with Nakayama algebras of finite global dimension (and some random algebras) gave no counterexamples.