So, I decided it myself. It looks like the problem is with NGINX HTTPUserIDModule (http://wiki.nginx.org/HttpUserIdModule). I used this to send the P3P header, but nginx did not send the header sequentially , possibly due to this line in its documentation:
The directive assigns a value to the P3P header that will be sent along with the cookie.
I saw how the headers were pounded when checking with a curled or trot, but not when I checked with Firebug or Charles. Perhaps there is a configuration option to fix this, but I could not find it.
In any case , if you pull the OUT header directive from the HTTPUserIdModule and just send the header manually by code, it works . This will probably work if you manually send the header using the HTTPHeadersModule http://wiki.nginx.org/HttpHeadersModule , but I have not tested this yet.
As soon as it worked through the code, I saw that I did not need a header for all the files - having it only in the file settings / receiving cookies was enough, contrary to many tips that I saw.
In addition, my CP was clearly too aggressive, despite the fact that the IBM policy editor said that everything is in order. I used a more basic CP and it worked.
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