Connected factors in image-free graphs containing an image-factor

A graph G is called image-free if G has no induced subgraph isomorphic to image. Let n, a, b be integers with image, image, and image. We prove that every connected image-free graph G has a connected image-factor if G contains an image-factor. This result is sharp in the sense that there exists a connected image-free graph which has an image-factor but no connected image-factor for image.