Let $x \in U$ be a germ of a $3$-dimensional terminal quotient singularity of type $\frac{1}{r} (1,a,r-a)$ over $\mathbb{C}$. Let $D \in |-K_U|$ be an anticanonical divisor. Assume that $D$ is normal and singular at $x$. Let $\mu: \tilde{U} \rightarrow (U,D)$ be a log resolution such that $\mu$ is isomorphism outside $x$.
Question Is $H^1(\tilde{D}, \mathcal{O}_{\tilde{D}}(-\tilde{D}) =0$?
It seems that $H^1(\tilde{U}, \mathcal{O}_{\tilde{U}}(-\tilde{D}))=0$.
If $H^2(\tilde{U}, \mathcal{O}_{\tilde{U}}(-2\tilde{D})) =0$, the statement in the Question seems to be true. Is there a counterexample?
