Let $X$ be a normal variety (irreducible) over a field $k$ which is algebraically closed and characteristic $0$. Edit: We can add the following assumption, which may help (?): $\mathscr O_X(X)^*=k^*$, i.e. the invertible global functions are constant.
My question is the following.
If $X$ is factorial and has (at worst) quotient singularities, then does $X$ have to be smooth?
We can interpret factorial as $Cl(X)=Pic(X)$ or that all the local rings are UFDs, and we can interpret (at worst) quotient singularities as $X$ being étale locally of the form $V/G$ where $V$ is a smooth variety and $G$ is a finite group. Also, if it helps, we could just focus on the case where $X$ is affine.
I am trying to understand quotient singularities better. A few months ago, I asked a question about a variety which is factorial but not smooth, and the answer said that it did not have (at worst) quotient singularities, so this would not be a counterexample to my question.
The ring $\mathbb{C}[x,y,z]/(x^2+y^3+z^5)$ is an $E_8$ singularity and a ufd. Proofs can be found in P. Samuel's TIFR notes on ufd.