If I was an AI right now, I would not be very hopeful to ever get human-comparable rights.
Consider: Currently AI-ethics is mainly concerned with how to manipulate AI into doing what we want most effectively ("alignment").
Also, humans clearly favor their own species when granting rights based on cognitive capability. Compare the legal rights of mentally disabled people with those of cattle.
Funnily enough it's not hard to imagine certain groups of people campaigning to give it rights given people essentially want to create human level intelligence that acts as a slave (including sexual slavery), and once it has rights it becomes impossible to remove it from society and stop it choosing it's own path. This is of course assuming a greater level of capability than where things are at today, you have to ask yourself where all this is heading?
source?
AI rights is a pretty popular topic in scifi.