Upper bounds for GK-dimensions of finitely generated P.I. algebras

We prove that if $A$ is characteristic zero algebra generated by $k$ elements and satisfying a polynomial identity of degree $d$ then it has GK-dimension less than or equal to $k\lfloor d/2\rfloor ^2$. We conjecture that the stronger upper bound that the GK-dimension of $A$ is less than or equal to $(k-1)\lfloor d/2\rfloor ^2 +1$ and prove it in a number of special cases.