Hi. The certificate is a self signed certificate from a private server.

I think that is your problem, see https://metacpan.org/pod/IO::Socket::SSL#Common-Problems-with-SSL it says

Validation of self-signed certificate fails even if it is given with SSL_ca* argument.

The SSL_ca* arguments do not give a general trust store for arbitrary certificates but only specify a store for CA certificates which then can be used to verify other certificates. This especially means that certificates which are not a CA get simply ignored, notably self-signed certificates which do not also have the CA-flag set.

This behavior of OpenSSL differs from the more general trust-store concept which can be found in browsers and where it is possible to simply added arbitrary certificates (CA or not) as trusted.

So, one path towards a solution, figure out what CA-flag is and how to set it , openssl stuff

Or maybe you can get a free one care of mozilla Let's Encrypt - Free SSL/TLS Certificates