Suppose the random variable $X$ is distributed over the real line. There are two cumulative distribution functions (CDFs) $F_X$ and $G_X$. The expected values of $X$ exist and are finite under both $F_X$ and $G_X$. Furthermore, suppose $F_X$ first-order stochastically dominates (FOSD) $G_X$, that is, $F_X(x) \leq G_X(x) \forall x \in \mathbb{R}$ and $F_X(x_0) < G_X(x_0) $ for at least one $x_0 \in \mathbb{R}$.
Suppose there is another random varibable $Y$. $Y$ and $X$ are dependent random variables - not merely deterministic functions of each other. Thus, $F_{X,Y}$ and $F_{X|Y}$ represent the joint and conditional CDFs following the marginal $F_X$, respectively. The same applies to the joint and conditional CDFs following the marginal $G_X$: $G_{X,Y}$ and $G_{X|Y}$. Again, all expected values (joint, marignal, and conditional) exist and are finite.
Suppose that the distributions conditional on $X$ are identical, that is, $F_{Y|X}(y|x)=G_{Y|X}(y|x) \forall x \in \mathbb{R}$. Can we conclude from FOSD of $F_X$ over $G_X$ and the previous sentence that $F_{X|Y}$ FOSD $G_{X|Y}$?