Let $k$ be a field, and let $A$ be a commutative $k$-algebra which is noetherian.

Suppose that for each prime ideal $p$ of $A$, it holds that the field $k(p)$, the field of fractions of $A/p$ has finite transcendence degree over $k$.

Does this imply that $A$ is a localization of a finite type $k$-algebra?