Let $(R,\mathfrak m)$ be a Noetherian local domain of dimension at least $2$. Then, must there exist $x\in \mathfrak m \setminus \mathfrak m^2$ such that $xR$ is a prime ideal of $R$? What if we also assume $R$ is normal?
My thoughts: If $R$ is a UFD, then every height $1$ prime ideal is principal, and in that case the problem boils down to finding a height $1$ prime ideal not contained in $\mathfrak m^2$, can this be always done? Outside the UFD case, I have no idea.