As measured by the IOS app store.
To resolve, I will first look at the Apple's list of top iphone apps. If that is unavailable, I will look at number of downloads. If that too is unavailable, I will consult the number of ratings. If all three of those are no longer public, or if the app store isn't a thing anymore, I will consult the community and my best judgement on what the biggest app is.
The prospect of sex or romance must be the primary marketing for these apps. Chatgpt and Character.AI do not count as romance apps, nor do Facebook and Whatsapp. Hinger, Tindr and Grindr are human romance apps. Nomi and Candy AI are AI romance apps. The biggest app does not necessarily have to one of the apps I named, but these examples should elucidate the boundaries of a "romance app".
What do people look for in connection and romance? What they feel has as much to do with what they believe the partner feels. If most people don't believe the AI has it's own experience dating AIs will remain a minority interest. This question therefore strongly correlates with "will most people think AI has a mind my July 1 2028", to which the answer is: decidedly not; though I could be wrong, some already do think that.