Well to date when this has come up it’s been the company that put the AI in from of their customers that’s held liable in the first instance. See Air Canada case where an AI made up non existent policies that courts ruled Air Canada even had to honor.
Naturally then the company could turn around and sue the maker of the AI for damages and this is the bit that’s less tested to date.
But it’s clear the answer isn’t “nobody” is liable. So if this AI marketing widget makes up false claims about a product that’s going to be a big problem for companies that put an AI in front of their customers.