Mobile software engineering has been a hot research topic for decades. Our fellow researchers have proposed various approaches (with over 7,000 publications for Android alone) in this field that essentially contributed to the great success of the current mobile ecosystem. Existing research efforts mainly focus on popular mobile platforms, namely Android and iOS. OpenHarmony, a newly open sourced mobile platform, has rarely been considered, although it is the one requiring the most attention as OpenHarmony is expected to occupy one-third of the market in China (if not in the world). To fill the gap, we present to the mobile software engineering community a research roadmap for encouraging our fellow researchers to contribute promising approaches to OpenHarmony. Specifically, we start by presenting a tertiary study of mobile software engineering, attempting to understand what problems have been targeted by the mobile community and how they have been resolved. We then summarize the existing (limited) achievements of OpenHarmony and subsequently highlight the research gap between Android/iOS and OpenHarmony. This research gap eventually helps in forming the roadmap for conducting software engineering research for OpenHarmony.