Under COPPA, if a person says their age on a site and apparently were younger than 13 when they joined, does the site have to do anything?

Question

If a user on a website with a 13+ age rule as required by The Children's Online Privacy Protection Act, such as Stack Exchange, signs up in year 1, then says in year 3 that they are less than 15 (and thus were less than 13 when they signed up), does the company running the site legally have to do anything, such as deleting the data from before they were 13?

_{I know that this law is de facto almost powerless; I'm just curious about how it's supposed to work.}