The same ideal in different polynomial rings.

Question

Let $J\subseteq k[x_1, \ldots, x_{n-1}]\subseteq k[x_1, \ldots, x_n]$ be an ideal such that $(x_1, \ldots, x_{n-1})$ is a minimal prime of $J$ (thinking in the polynomial ring $k[x_1, \ldots, x_n]$). Is $J$ an ideal $(x_1, \ldots, x_{n-1})$-primary (thinking in $k[x_1, \ldots, x_{n-1}]$)?