Thanks, Iosif. Sure, please close this question with your counterexample.

Thanks Kodlu! Your idea of using Polya's Enumeration Theorem sounds like a right track. However, I still didn't see how it applies to my problem. PET can count the number of colorings up to rotation and/or reflection. In my problem, we are counting the number of colorings subject to the constraints f(x)=l, and such constraint seems not equivalent to either rotation or reflection. It is clear that if x is a rotation or reflection of y then f(x)=f(y), but not vice versa. PET still sounds promising and it is just unclear how to view the set {x: f(x)=l} as an equivalent class. Sorry if I miss sth.

Thanks, Terry! To make your arguments complete: suppose $P(E(X|\mathcal{A})>1)>0$, then there exists an $\epsilon>0$ such that $A_\epsilon=\{E(X|\mathcal{A})\ge1+\epsilon\}$ has positive probability. Then $$(1+\epsilon)P(A_\epsilon)\le\int E(X|\mathcal{A})I_{A_\epsilon}dP=E(XI_{A_\epsilon})=E(X|A_\epsilon)P(A_\epsilon)\le P(A_\epsilon)$$, contradictiion!

I believe (***) is right. Please see mathoverflow.net/questions/238805/…

Yes, Rio's results might be applied. I am adding my tentative solution soon...