## rt.representation theory – Limits for the finitist dimension

The finitistic dimension of an algebra is defined as the supreme of all the projective dimensions of the modules that have a finite projective dimension.

For finite dimension algebras. $$A$$ with the radical zero cube it is known that for the finitist dimension $$fd (A)$$ one has $$fd (A) leq dim (A) ^ 2 + 1$$ (See https://www.sciencedirect.com/science/article/pii/S0021869383712056).

Question: we have $$fd (A) leq dim (A) ^ {r-1} + 1$$ for algebras $$A$$ with $$J ^ r = 0$$ when $$J$$ is Jacobson's radical of $$A$$ Y $$r geq 3$$?

(Of course, one can probably only expect a counterexample in an answer here)