There are no prominent 'cage open visual novel' titles that come to mind. Visual novels usually gain popularity based on various factors like their art style, story, and characters. Since 'cage open visual novel' is not a well - known genre - specific term, it's unlikely that there are popular titles directly associated with it. But in the vast world of visual novels, there could be some hidden gems with a related concept that haven't gained wide recognition yet.
As far as I know, there are no widely known visual novels with the exact title 'cage open visual novel'. However, there might be some indie or niche visual novels that could potentially use a similar concept in their stories. But they might not be mainstream enough to be considered popular with a broad audience.
I'm not aware of any specifically popular 'cage open visual novel' titles as this seems like a rather unique or less common concept. There are thousands of visual novels out there, but not many that I know of would be named exactly this way.