For a Noetherian, separated integral scheme $X$ that is regular in codimension $1$, let $Cl(X)$ denote it's (Weil) divisor class group. For such an affine- scheme $X=\operatorname{Spec}(R)$, let us write $Cl(R):=Cl(\operatorname{Spec}(R))$.
Now I know that if $R$ is a normal domain and $Cl(R)=0$ then $R$ is a UFD. But I'm not sure if one can remove the hypothesis "normal". So here's my question:
Let $(R, \mathfrak m)$ be a Noetherian complete ($\mathfrak m$-adically) local domain of depth $2$ and dimension $3$ such that $R_P$ is regular for every height $1$ prime ideal $P$ of $R$ (i.e. $R$ satisfies $(R_1)$). If $Cl(R)=0$, then is $R$ necessarily a UFD (or equivalently, is $R$ necessarily normal) ?
If there is a counter-example, then it must be non Cohen-Macaulay, because Cohen-Macaulay local ring of depth $\ge 2$ implies $S_2$ so along with the $R_1$ assumption, the ring would be normal, hence $R$ would be a UFD if the divisor class group is trivial.