There are image-uniform cubefree binary morphisms for all image

A word is cubefree if it contains no non-empty subword of the form image. A morphism image is k-uniform if image has length k for all image. A morphism is cubefree if it maps cubefree words to cubefree words. We show that for all image there exists a k-uniform cubefree binary morphism. By a result of Leconte, this implies the following stronger result: for all image and image, there exists a k-uniform image-power-free binary morphism.