How to send requests from LWP through socket tunnel returned by Net::OpenSSH?

Question

Perl

There is a code like this:

#!/usr/bin/env perl

use utf8;
use strict;
use warnings;
use feature qw/say switch unicode_strings/;

use Data::Dumper;
use Net::OpenSSH;
use LWP::UserAgent;

my $ssh = Net::OpenSSH->new("192.168.1.1", user => 'user', password => 'PaSsWoRd' );
$ssh->error and die "Couldn't establish SSH connection: ". $ssh->error;

my ($socket, $pid) = $ssh->open_tunnel('test.com', 80);
say "Socket[$socket] PID[$pid]";

my $ua = LWP::UserAgent->new;

How to force the same $socket to become a socket for connections from LWP?
The options to set up a local Sox proxy server using an ssh client and run LWP through this proxy server as through regular Sox do not work.