This issue comes up quite regularly.  See the thread WWW::Mechanize with https and a proxy  (in particular the linked AnnoCPAN note).

The upshot of that is to set

$mech->proxy(https => undef);
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and not use $mech->env_proxy() — at least not after having called $mech->proxy(https => undef).   env_proxy() is being called from the WWW::Mechanize constructor anyway, but explicitly undefining it afterwards overrides any settings read from the environment.

The underlying SSLeay library will in this case read the proxy settings from the environment itself.  See Proxy Support.

Update: just verified that even with the most recent version 5.834 of LWP::UserAgent (that WWW::Mechanize inherits the env_proxy() method from), this is still the same old issue.