Uniqueness of constant mean curvature surfaces properly immersed in a slab

We study complete properly immersed surfaces contained in a slab of a warped product $\mathbb{R}\times_\varrho\mathbb{P}^2$, where $\mathbb{P}^2$ is complete with nonnegative Gaussian curvature. Under certain restrictions on the mean curvature of the surface we show that such an immersion does not exists or must be a leaf of the trivial totally umbilical foliation $t \in \mathbb{R}\mapsto \{t\} \times \mathbb{P}^2$