Convergence is mostly a software thing. There are two parts to this:
flexibility: the software including shell and application GUI adapts to the capabilities of the hardware it runs on.
This means that the same software, with the same features, works well on a desktop, a phone or a toaster.
agility: the software including shell and application GUI adapts to changes to the capabilities of the hardware is running on.
This means that connecting a display, keyboard, dock etc. will enable user interactions optimized for these.
Most existing software is not designed for either of these goals. The Ubuntu Touch experience of using "desktop" apps in Libertine speaks to this, it is possible to use these applications, but the interaction is poor because these apps do not adapt to the hardware. This can change but, like any change, there will be costs, pain and resistance until suddenly it will be done and everyone wonders why it was ever any different.
But the software is only a part of what is being alluded to here. Real use-cases are not about using a specific piece of software, they are about accomplishing goals: publishing something, learning something, meeting someone. That needs a combination of user data, software and access to hardware.
I've had a long career in software development and seen computing shift from centralized "mainframe" computers that hold the data and programs for multiple users to "personal" computers that hold a users data and programs and back to centralized "clouds" that hold all the data. That change has been driven by economics: changes in the costs of computers and of connecting them.
While there are legitimate privacy and security concerns the economic drivers are currently pushing towards centralization and connections. Many apps are little more than optimized front-ends for data and computational services elsewhere on the internet. I think this trend will continue before a readjustment back towards "personal" computation.
In this context, any "one device" we might carry is more about having the keys (or passwords) we need to accomplish our daily tasks than carrying all the software and data we use. If both my laptop and phone can support all my daily tasks interchangably, browsing the web, emailing, Telegram, IRC, software development, video calling, phone calls, SMS, etc. then I need only take the one best suited to the occasion. Having the same software and keys on both is what would make them interchangable. (And yes, phone calls are best on a phone while software development is best on a laptop, but making both possible gives me control.)