As a sort of a follow-up and a generalization from a previous question, suppose that we have two independent, identically distributed random variables $X, Y$ and a third random variable $W$. Is it true that $\mathbb E[X \mid W]=\mathbb E[Y \mid W]$?
More in general, what conditions are to be imposed in order to have equality? When $W=X+Y$ the statement is true. But supposed there is no relation, a priori between $W$ and $X, Y$. What can we say in this case?
Suppose $W = X$. Then $E[X\mid W] = X$ whereas $E[Y\mid W] = E[Y]$. There is too much freedom in the choice of $W$. You have to control for some property to arrive at a useful conclusion. Otherwise, any conclusion is possible as the example above shows.