Push Notifications for New Chatbox Messages

As you know, we have Push Notifications thanks to the PWA (Launched some months ago) and i'm thinking that if we have push notifications of new answers, private messages and more, why not when a member sends a new message in the Chatbox, receive too a push notification like

New Message in the Chatbox

Or something like that.
And yes, i know about the tutorials to get notified by a new sound or toolbar notification, but if we can see the push in browser, could be great.

What's your oppinion?
Let me know

I voted Yes

I have voted yes, assuming that, of course, we get the option to disable these pushnots.