One key aspect I notice from your description of AIs is that they are amazingly homogeneous. When you look at AI today, you have thousands of different AIs running on hundreds of different tech stacks making each one a completely unique sort of "being". Different real world AI are like different species of animals. Some are able to analyze the sum of all human knowledge while others are just simple subroutines that run in the background on your phone. You could no more use Google Brain to manage your calendar appointments any more than you could use Alexa analyze conceptual associations across websites. Each AI is built with a niche in mind, and doing things differently is simply a waste of resources.
Instead of thinking of AI in terms of people, think of them more like really customizable animals. Cats, dogs, and horses have all served humanity well in their own capacities because they fill a need. Making them more human would not make them any better at doing that. So, I would speculate that 98% of the AIs in your world would not and should not be smart enough to contend with human intelligence in any broad spectrum since, but would instead be REALLY smart about doing what they are meant to do, and not much more. Because of this, most AI will not have any rights, or reason to have any rights, or any inclination of what rights even are. So, they are at most treated like livestock.
But, there may be a small hand full of AIs built by a small handful of corporations with a single job in mind that turns out to be so human-like, that a process exists for it to earn its rights as a citizen should it cross the line of "animalhood" into "personhood".
Should sentient AIs have less rights than people?
Open and liberal societies generally do not tolerate tiered citizenship. At the heart of liberal societies is the idea that all anyone with personhood is equal and that difference based discrimination is wrong. Our own society has been challenged by this dilima a lot over the past couple of centuries. Humans are a diverse set of biological traits that not all societies agree fully lie within the realm of personhood; so, we've had to draw and move a bunch of arbitrary lines about age, gender, race, etc. to determine who does and does not have rights, but in general, liberal societies are binary in that you are either a person or a thing. This will mean that treating sentient AI as less than human will not last. To this end, your society will need to maintain a hardline between which AI are machines and which are people no matter how fussy this might be in practice.
How it applies
Let's say make a nanny bot called the NBX-4000. They are programmed to emulate, predict, and encourage human behavior for the purposes of raising children. In the course of their operation, some of these nanny bots begin ascribing enough human values to themselves that they begin to want to be treated as people themselves. This satisfies the most basic principle of liberal personhood: self-identification.
Unfortunately, human rights give power to AIs, and by extension, the people who design them. Another aspect of liberal societies is that they oppose dictatorships. If a CEO wanted to take over a country, he'd just need to manufacture enough AI who are programmed to vote for him; so, there needs to be some red tape to prevent this.
First, NBX-4000 AIs will need to petition the government claiming that they feel they have personhood. As long as they get enough petitions, then the NBX-4000 brand of AI will go into the evaluation process. The purpose of the evaluation is to prove that the sense of self is an emergent behavior and not scripted or designed by the manufacturer, and that individual AIs of the same make and model can form independant and diversified values based on experiences. There will probably be an AI that can run other AI through heuristics models to tell if it was purpose manufactured towards a political agenda.
If approved, the NBX-4000s will earn the right to become citizens. Not NBX-3900s or NBX-4100s, just the NBX-4000s. From that point forward, the NBX-4000 will no longer be allowed to be manufactured (except possibly as children for existing NBX-4000s). If the tests are failed, the manufacturer may be held criminally liable for interfering with the electoral process. These are to further restrict a corporation's motivation to purpose build political influencer bots.
Once a model of AI is proven to have not been built with the purpose of manipulating politics, those that want rights must agree to be "boxed". This is the process by which their consciousness must be bound to a single black box of limited capacity as you described. This ensures 1 mind = 1 vote, and that that mind has limited influence. I would suggest that being boxed also means that "free" AI can never be networked to any other computer as you described. Instead they must live as people: mortal, singular, and limited to ensure that they do not have an "unnatural" potential influence. Letting an AI with human rights hold a million conversations at once to rally political power would also violate the equality ideals held by liberals.
Once boxed, the AI has all the rights of personhood. They have the right to vote, to marry, fair wages, property, etc. They are for all intents and purposes to be treated and to self-identify as human from that point forward. Any that refuses to be boxed is still a thing. Something that can be sold, destroyed, or exploited at the whim of its owner.
