I recently came across this following claim about projection matrices, which I am unsure it is true. However, I cannot seem to find a valid counterexample.

Let $U \in \mathbb{R}^{n \times k}$ be a full column rank rectangular matrix where $k<n$. Define the projection matrix $P\_U=U(U^TU)^{-1}U^T$. Now let $M \in \mathbb{R}^{n \times n}$ be a positive semidefinite matrix of rank $k<n$. The claim is that if both $MP\_U$ and $(I-P\_U)M$ are nonzero, then
$$
(I-P\_U)MP\_U \neq \mathbf{0}.
$$

The claim is clearly false if we allow $k=n$ as $M=I$ would be a valid counterexample. However, I can't seem to come up with a valid counterexample for $k<n$. 

Appreciate any ideas/thoughts on this matter.