There is an opinion that Exhentai remembers the IP from which the account is logged in and does not allow you to log in from two devices at the same time. Perhaps this assumption is wrong, but it was by reading that discussion that I understood how to overcome the blank screen.
How I did it:
1) You will need an account at e-hentai.org. My account is over a year old, so I can't really say how important the account lifespan and Moderation Power is. This account has a moderator power of 6.
2) You need a browser extension to manage/edit cookies. There are quite a lot of them.
2.1) You may need a VPN whose IP address was not a source of massive traffic to e-hentai and exhentai. (I was able to find a little-known public VPN with an IP in the USA, it worked. )
3) Login to e-hentai.
4) Open exhentai. (I just changed the address bar in the same tab.)
5) Most likely at this moment you will see a blank screen. This is normal, open the Exhentai cookies in the extension to edit cookies.
6) Look at the value in the "igneous" line.
6.1) If the value in this line is a set of numbers and letters, then everything is fine. Using the extension, remove the "yay" line from the cookie, and only that, and then reload the page. At this moment you should be allowed into the exhentai website.
6.2) If the value is "mystery", then it didn’t work out, close the exhentai website or return to the previous page and delete exhentai cookies completely.
7) At this point, you can try to change the VPN and then open exhentai again. If the value in the line “igneous” is “mystery”, then repeat steps 6.2 and 7. If the value in the “igneous” line is indicated by numbers and letters, then go to point 6.1.
In addition, after you have received working cookies, you will be able to log in through regular public VPNs, although there is a chance that their IP will be banned for a massive download, there will be no blank screen.
Also, it seems that if you manually transfer three lines of working cookies to another browser or to another device, you can avoid repeating this entire process. Although I haven't checked this yet. And, naturally, this method should only work for the same account.
This is how I logged into exhentai. I hope this helps someone.
Upd. I accidentally discovered that I don't need a VPN for exhentai. There are no blocks on it, unlike e-hentai.
Upd2. I tested everything on the latest version of Firefox on Android without all the tweaks that I did with the main browser and I really don’t need a VPN. What a strange feeling.
I also checked that transferring the working value “igneous” from browser to browser on one device and removing “yay” allows you to log in calmly. This works even better than I hoped; the remaining two lines are automatically copied from the e-hentai cookie and do not need to be copied manually.
Good luck, men of culture.