Question on contact metric manifolds which Reeb vector field isn't an eigenvector of the Ricci operator?

Question

In here I asked a question with no answer. In this topic I have another question:

Question: Can anybody construct an example of 5 dimensional contact metric manifold $(M,\eta,\xi,g,\varphi)$ such that $Q\xi\neq\lambda\xi$, where $Q$ is Ricci operator and $\lambda$ is a smooth function on $M$. In 3 dimensional case I found such example.