Nginx SNI + OCSP stapling not working

I recently tried to setup OCSP on one of my nginx servers.
Unfortunately I couldn't get it to work and didn't find a solution so far.

The configuration looks like this:

ssl_certificate /etc/ssl/private/;
ssl_certificate_key /etc/ssl/private/;
ssl_trusted_certificate /etc/ssl/private/;

ssl_stapling on;
ssl_stapling_verify on;
resolver valid=300s;
resolver_timeout 15s;

The fullchain.crt contains the servers cert, the intermediate and the root cert.

If I check these certs by hand with :

openssl ocsp -issuer intermediate.crt -CAfile fullchain.crt -cert cert.crt -url -no_nonce

it returns ok:

Response verify OK
cert.crt: good
    This Update: Apr  7 11:26:10 2018 GMT
    Next Update: Apr 14 11:26:10 2018 GMT

But checking the server with s_client from elsewhere always returns

OCSP response: no response sent

even after waiting several minutes and nginx always throws the error:

2018/04/09 12:59:06 [error] 9474#9474: OCSP_basic_verify() failed (SSL: error:27069065:OCSP routines:OCSP_basic_verify:certificate verify error:Verify error:unable to get issuer certificate) while requesting certificate status, responder:

The server uses SNI since it delivers multiple sites with different certificates.
Somebody got an idea what I am missing here?