When you open chat, you can also open people.
By clicking the people button.
But when you click the chat button again to close chat.
It's not closing people.
My idea is that the people window is also closing with the chat window. But remember the last state of the people window. So when the people window is open and you close chat. That it knows it need to open people again next time you open chat.
Other option, It could be option to combine both chat and people into one window that can be a bit higher then. and put chat and people in separate tab.
I have not seen much use cases to have both open on the same time. a tab would do the same.