A PI degree theorem for quantum deformations

Let $F$ be an algebraically closed field. We show that if a quantum formal deformation $A$ of a commutative domain $A_0$ over $F$ is a PI algebra, then $A$ is commutative if ${\rm char}(F)=0$, and has PI degree a power of $p$ if ${\rm char}(F)=p>0$. This implies the same result for filtered deformations (i.e., filtered algebras $A$ such that ${\rm gr}(A)=A_0$).