Suppose I have two random variables $X$ and $Y$ that take both positive and negative values such that $\mathbb{E}[X]=\mathbb{E}[Y]=0$, and the absolute value of $X$, i.e., $|X|$, first-order stochastically dominates the absolute value of $Y$, i.e., $|Y|$.
Does this imply that $X$ second-order dominates $Y$? If not, are there additional conditions (e.g., symmetry?) such that second-order dominance is implied? Any references would be greatly appreciated!