# Bounds for the finitistic dimension

The finitistic dimension of an algebra is defined as the supremum of all projective dimensions of modules having finite projective dimension.

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

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

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