IntelliJ IDE with GitHub Enterprise and Cookies - no access.

Hi! 
I am using IntelliJ in my work every day, but I can't use its full power because we have an additional layer of protection in my company.
To access GitHub Enterprise we need to use token (generated in the GHE once) and global Cookie (generated by a special internal tool every 24 hours)
For all GIT operations, IntelliJ works perfectly also with token+cookie.
But I can't set up full GitHub support in the settings by adding a new GitHub Enterprise profile to Version Control\GitHub settings - I have auth error.
If we switch off Cookie requirements in the company - all start to work fully and without any problems.
But right after switching back Cookie requirements - IntelliJ stops to work.
Looks like IntelliJ can't use global Cookies when accessing GHE.

Can you advise me on how to force IntelliJ to use stored in the global Cookies token for accessing GHE?

Maybe it will help: mod_auth_openidc_state - the name of the property in the Cookie with Token 

*By the way - this is a really-really huge international company and everyone has this problem with GHE and global Cookie with token.