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)


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