Really just depends on what you consider convergence. I'd love to see programs via appimage, flatpak, or even snap that could just be installed on one platform or another. The problem is all these platforms are different and the software developed on them are different as well with different goals. The closest I've found to a convergence system that works is Chrome OS and Android.
Essentially you would want software that is scale-able regardless the platform. Have a program that you can install on the phone, then turn around and be able to install on the desktop and it scale up. The closest I've seen to something like this is the maui framework under Nitrux.
This way the program being used is the same regardless the platform and makes everything feel seamless. I've also seen Chrome OS essentially do this with Android applications being run on their desktop OS. The only real difference in running an application on Android and Chrome OS (aside from the underbelly making things work on desktop) is you have a minimize, maximize and close slapped onto the application. It would be amazing for eample to see the open store ported to desktop and all applications just work.
Hardware to me is kinda the beauty of Linux. 2 or more gigs of ram minimum and for most people you are set. Usage for a wide majority of people anymore consists of paying bills online, word processing, and surfing the web or multimedia.
As for OS itself the closest I've seen to convergence is Android x86. Unity8 I could see doing the same as long as it's essentially scale-able. As of current it's not really like that. The phone OS and Desktop OS for nearly every major system out there is difference in appearance.
For me either make it all function and/or look as close to the same on all platforms and slap on a minimize, maximize, and close for the desktop and I'm set.