  We give a condition which ensures that the Paneitz operator of an embedded
three-dimensional CR manifold is nonnegative and has kernel consisting only of
the CR pluriharmonic functions. Our condition requires uniform positivity of
the Webster scalar curvature and the stability of the CR pluriharmonic
functions for a real analytic deformation. As an application, we show that the
real ellipsoids in $\mathbb{C}^2$ are such that the CR Paneitz operator is
nonnegative with kernel consisting only of the CR pluriharmonic functions.
