A question on conditional expectation leading to zero covariance and vice versa

In my probability class I was tackled with this seemingly weird question involving conditional expectation:

Let X,Y be two random variables (it is not mentioned whether or not they are discrete or continuous) and we are asked the following:

  1. For all constants $ \beta $ we have $ E[X | Y = \beta] = E[X] $

  2. $ cov(X,Y) = E[XY]-E[X]E[Y] = 0 $

The problem: we are asked to prove or give a counterexample to 1 leads to 2 and to 2 leads to 1

I have tried to prove these two directions but got nothing just by looking at the definitions and so I thought maybe they are false and we are to give counterexamples but I got nothing there either. Looking at the covariance formula I see Y is involved but I cannot really seem to incorporate it from conditional expectation given, so I am stuck and need help. Thanks to all.

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