Level set flow in 3D steady gradient Ricci solitons

Let $(M^3,g,f)$ be a nontrivial 3-dimensional steady gradient Ricci soliton. If the scalar curvature $R$ satisfies $c_1{r}^{-b}\le R\le c_2{r}^{-a}$ for some $a\in (0,1],b\ge a$, and $c_1,c_2>0$, then the umbilical ratio of the level sets of $f$ satisfies $\frac{2|A|^2-H^2}{H^2}\in O(r^{6a-\frac{8a^2}{b}})\cap O(r^{2b-4a})$.