Let $k$ be an algebraically closed field of characteristic zero, let $G$ be a connected reductive linear algebraic group over $k$, and let $P$ be a parabolic subgroup of $G$. So we have the flag variety $G/P$. Suppose that this flag variety $G/P$ is not a toric variety, or equivalently that $G/P$ is not a product of projective spaces, or equivalently that the Cox ring of $G/P$ is not a polynomial ring over $k$.
I want to look at the spectrum of the Cox ring of $G/P$ in this case. As mentioned in Brion's paper "The total coordinate ring of a wonderful variety", the Cox ring of $G/P$ is equal to the coordinate ring of the quasi-affine variety $G/[P,P]$. Let $Y$ be the affine variety which is the spectrum of the coordinate ring of $G/[P,P]$. Again, by our supposition, $Y$ is not affine space.
In this case, does $Y$ have worse than quotient singularities?
Clearly $Y$ is smooth if $G/P$ is a toric variety (in this case, $Y$ is affine space), which is why I want to rule out this case. My hunch is that, aside from this case that we have ruled out, $Y$ must have worse than quotient singularities.
In the question that I asked here, the answer shows that, in the case of $G=SL_3$ and $P=B$ the Borel subgroup, the corresponding $Y$ indeed has worse than quotient singularities (in this case, $[P,P]=U$ is the maximal unipotent subgroup). For clarity, $Y$ in this example is the spectrum of $k[u,v,w,x,y,z]/\langle ux+vy+wz\rangle$.