What launch strategy would you suggest for product Hunt?

I'm in the process of preparing my product launch and am wondering what the best strategy would be for Product Hunt. FYI, there are currently 1000+ beta testers on a waiting list + about 500 personal relations I can contact. My product is a productivity app (a Chrome extension for Gmail) that helps you highlight what's important in the emails you receive or send, and also prepares a quick response template that only requires you to type your responses to key phrases. So far, no one has used the product yet, as we are in the process of completing the development. So I'm looking to build a first community that will help me debug the product and suggest improvements in a private beta, then open the beta to the public, and finally exit the beta. Do you think it's a good idea to set up a launch on PH at this private beta stage, or should it be set up only at the public beta stage? Thanks for sharing your opinion and advice 🙏