On nonrational divisors over non-Gorenstein terminal singularities

Let $(X,o)$ be a germ of a $3$-dimensional terminal singularity of index $m\geq2$. If $(X,o)$ has type $cAx/4$, $cD/3\text{-}3$, $cD/2\text{-}2$, or $cE/2$, then we assume that the standard equation of $X$ in $\mathbb{C}^4/\mathbb{Z}_m$ is nondegenerate with respect to its Newton diagram. Let $\pi\colon Y\to X$ be a resolution. We show that there are at most 2 nonrational divisors $E_i$, $i=1,2$, on $Y$ such that $\pi(E_i)=o$ and the discrepancy $a(E_i,X)$ is at most 1. When such divisors exist, we describe them as exceptional divisors of certain blowups of $(X,o)$ and study their birational type.