Assume two random variables $X$ and $Y$. It is given that: $$E\left( X \right) \leq E\left( Y \right)$$
Now we have another random variable $Z$. What can be said about the relation between $E\left( {X|Z} \right)$ and $E\left( {Y|Z} \right)$? It is also known that $X,Y,Z$ are strictly positive.
It is clear that ${E_z}\left( {E\left( {X|Z} \right)} \right) \leq {E_z}\left( {E\left( {Y|Z} \right)} \right)$ (using the law of total expectation), but I am interested in the relation between the conditional expectations, rather than between their expected values.
Assume that $Z$ is uniform on $(1,3)$ and that $X=Z$ and $Y=4-Z$. Then $E[X]=E[Y]=2$ while $E[X|Z]=X$ and $E[Y|Z]=Y$ hence $P[E[X|Z]>E[Y|Z]]=P[E[X|Z]<E[Y|Z]]=\frac12$. In other words, neither $E[X|Z]>E[Y|Z]$ nor $P[E[X|Z]<E[Y|Z]$ almost surely.