How to make form fields work inside Dialog?

[wiz256]

Hello, what are the options to make form fields work inside Dialog?

I have a repo to demonstrate the problem on the first two inputs on the page (name and birthday):
https://codesandbox.io/p/github/wiz256/conform-to-shadcdn-field-in-dialog-problem

The real reason is that on small screens like mobile, where keyboard takes a lot of screen estate, I would open the input (like ComboBox with auto-suggest and date selector) inside a "full screen dialog" making sure the input goes on the very top of mobile screen.

How to make such input fields from inside dialog be in sync with the form data and behave like it should?
I tried different approaches but nothing helped.