Non-isomorphic algebras with equal Hilbert-Poincaré series

Question

Let $A,B$ be two finite-dimensional graded algebras and let $P_A(x),P_B(z)$ be theirs Poincaré series. Suppose now that $P_A(x)=P_B(z)$.

Question. Is it implies that $A \cong B?$

Answer 1

Hint. $A=\mathbb R[x,y]/(x^2,y^2)$ and $B=\mathbb R[x,y]/(xy,x^2-y^2)$ have the same Hilbert (Poincare) series, but they are not isomorphic.