Early Access - Cross-Platform?

Hello everyone.
Because I haven't played PoE1, I created an account for the PoE2 Early Access and linked it to my Microsoft/XBOX account.
Since I want to play PoE2 on both XBOX and PC, I bought and installed the game in the XBOX store.
The game works perfectly on XBOX.

When I go to my account, I can see that my Microsoft/XBOX account is linked to my GGG account and I have received the 300 shop points.
However, when I go to Early Access, the page tells me that I don't have access to Early Access and need to buy it.
I also can't log in to the PC client.

Do I need to buy Early Access again to play on PC?

Greetings.