Finding a small prime ideal above an ideal

Question

Let $I$ be an ideal of $\mathbb C[x_1,\dots,x_n]$. Define the $g$-degree of $I$ as the minimal transcendence degree (over $\mathbb C$) of a field generated by a generating subset of $I$.

Let $I$ be any ideal of $g$-degree $k$.

Is $I$ contained in a prime ideal of $g$-degree $k$?

yes if $I$ is 1-generated.