Ajax Request from incognito mode

I have an extension and a site.
My extension makes an ajax request on click to my site and get a new session id. Then with this SessID redirects to my site.
In normal mode everything works well, but not in incognito mode.
It seems that ajax request is made in normal mode and the site page opened in incognito mode. Of course, SessIDs will be different.

There is an option in the manifest file for chrome-based browsers: “incognito”:“split”. But it doesn’t support in Mozilla WebExtensions - only “incognito”:“spanning”.

Now, how can I make an incognito ajax request from my extension to my site in incognito mode? Is it possible?