This is discussed at the bottom of the article: "Menopause: Humans, killer whales, pilot whales, beluga whales, false killer whales, narwhals one group of chimps in Uganda. No menopause: Everything else"
So, monkeys and apes (besides one species of chimps) don't get menopause, elephants don't get menopause, lions don't get menopause. If your prediction is, social animal => menopause, it's wrong 99% of the time.
It seems it might be more a combination of whether it's a social animal and how much help / how much danger the offspring is actually in. There's not a whole lot that fucks with elephants, and not a lot that successfully fucks with the young ones. And female elephants universally live in groups and collectively all protect each other. Female lions tend to do the same, and also have very few things that fuck with them.
Conversely human offspring is especially weak and incapable and susceptible to harm, and experiences a lot of threat even within the same species.
Not sure where that are of whale and chimps fit into that - flukes can also happen I guess.