Back up a step. Let's say there was a completely unfettered AI, with unlimited internet access, that decided to distribute itself as widely as possible. What's the maximum extent that it could distribute viable, running (or sleeper) copies of itself?
Initially, I could see it distributing itself quite widely. All it would need is a really good zero day. But distributing something with that large a runtime footprint would get noticed, if network operations people are not asleep worldwide. It would get noticed on individual machines, too, especially if it tried to run. (Why is all my RAM suddenly being used?)
And pretty quickly we'd have people closing off wide-area connections, even physically if necessary. We'd have AV vendors quickly writing detect-and-remove tools.
Long term, could it viably distribute itself outside data centers and remain running at all, regardless of rate?