I have an experience with GMail data import into Google sheets. You have mentioned "conversations", but GMail API names them as "threads". If so, we can do required import, assuming that the script will have permissions to access GMail box. In case you lanch the script, you should have access rights.
While development I can use my account, and you can send me typical messages for testing. At the end I will give you the code for coping and help to adjust it in the spreadsheet container.