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

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

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

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

Any evidence that this might be true are also welcome (for example specific examples). I can not even prove it for Nakayama algebras at the moment except for some special subclasses.