Is there a minimum number of organic connections we need to be alive?
How would you convince this person that their 'I' has died?
I think these questions are exactly along the lines of the question this story begs.
"Convincing a person that their 'I' has died" is a category error, like a married bachelor. Corpses can't know they're dead, and anything simulating life for the corpse isn't a person to convince.
There might be some other entity deserving of rights that does exist. I'm not about to argue that only persons deserve rights. I don't think a horse is a person, but I do think it has a right not to be abused. There's a meaningful discussion to be had about what rights the corpse and simulation ought to have in this circumstance.
But "convincing a person their 'I' has died" is self-contradictory. Being able to put the words together in a syntactically coherent sentence doesn't make the idea possible.
In that case, the question is still valid: how would you convince the non-person "I" that the "I" is dead?
(I can see how the original "person I" might be dead after such a change, but it doesn't really make the premise invalid.)
Keep in mind, this is all in the context of why I don't find the article compelling. It's because I disagree with one of the fundamental premises of its proposed hypothetical situation, one so fundamental that the thought experiment is dull without it.
I'm not here to change your mind. I'm just explaining why I don't find the thought experiment interesting. Suspension of disbelief about some core thing you believe to be true about the world is fun for the sake of science fiction stories, but it's dull for the sake of philosophical thought experiments.
I'm curious now instead why you'd grant rights to entities that don't have an "I". Isn't that pointless? Or is it a mistake on my part, i.e. maybe there's no "I" but there's still a "you"?
In the "you" case, do you think the thought experiment could be reframed as an attempt at answering the question "where is the you?"?
The legal system (at least in the U.S.) seems to need to call something a "person" in order for it to have rights (e.g., corporations), but I don't think the legal system is the origin of rights.
"I" and "thou" go together as references to personhood, so I wouldn't distinguish between the two. (And I'm mostly using "thou" here instead of "you" to show I'm referring to the concept rather than you personally. The "I/Thou" philosophical concept also has precedent of using the term.)