Noetherian rings whose prime ideals have projective dimension bounded above

Question

For a module $M$ over a commutative Noetherian ring $R$, let $pd_R (M)$ denote the projective dimension of $M$ as an $R$-module. Now let $R$ be a commutative Noetherian ring such that $\sup \{ pd_R (Q) : Q \in Spec (R) \} < \infty$, then definitely $R$ is regular.

My question is: Does $R$ have finite global dimension ?