Consider the $3$-dimensional Noetherian local ring $R=\mathbb C[[x,y,z,w]]/(x^2-yz)$ with maximal ideal $\mathfrak m=(x,y,z,w)$ .
Then, is it true that $R_P$ is a UFD for every prime ideal $P\ne \mathfrak m$ and $R_Q$ is not regular for some height $2$ prime ideal $Q$ of $R$ ?
I can show that $R$ is normal, hence $R_P$ is regular for every height $1$ prime ideal $P$ of $R$. But I don't know what happens for localization at height $2$ prime ideals.
Please help.