According to https://superuser.com/questions/1314881/chrome-private-tabs-to-regular-tab
It should do what you want, ‘When you press the Incognito Tab Switch button, it opens up your current tab in an incognito window. Keep pushing the button, and each tab gets added to the new incognito session. If you enable the extension to work in incognito mode, you can also switch those tabs back to normal browser tabs with a click.’
Of course, you can simply copy the url from the normal tab, open a private tab and paste.