Can't connect to kuix.de:443 (10051)
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This means your computer cannot connect to the other computer.

Have you checked that your browser can connect to the website? Maybe there are proxy settings that you need to tell your Perl script?

Update: I now realize that you're already aware that you need a proxy. For SSL, you need to tell the SSL C libraries to use a proxy. The best approach is to set up the environment before these libraries are loaded. This is easiest done by setting up the environment before your Perl script is started. As an alternative, you can set up the environment in a BEGIN block before the SSL libraries are initialized:

BEGIN {
    $ENV{ https_proxy } = "http://my.proxy.example";
};
...
use LWP::UserAgent;
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