Height unmixed ideal

Question

Suppose $R$ is a regular local ring and $I$ is a non-zero ideal such that $I$ is a radical ideal and $I$ is height unmixed. Suppose $J$ is any radical ideal contained in $I$ and with the same height as $I$. Can we always find a prime $P\in \operatorname{Ass}(R/J)$ such that $P$ is not minimal.

Answer 1

$I=(x) \cap (y)$, $J = (x) \cap (y) \cap (z)$.