At the end of your code its too late. You're supposed to get more info about which Unknown Error it is. Try

#!/usr/bin/perl --

use strict;
use warnings;

use LWP::UserAgent 5.829;

@ARGV = qw[  http://www.eserve.com.sa/ ];

my $url       = shift or die "URL expected\n";
my $useragent = LWP::UserAgent->new;

$useragent->proxy(['http', 'ftp','https'], 'http://192.168.5.39:8080/'
+);
$useragent -> timeout( 20 );

my $request   = HTTP::Request->new( GET => $url );
my $response  = $useragent->request($request);
my $err =  sprintf "(%d)(%s)(%d)(%s)\n", $!,$!,$^E,$^E;
print $useragent->proxy('http'),"\n";
print "Err : $err\n";
print $response->dump,"\n";

__END__
lwp-request -useUS -m head http://www.eserve.com.sa/
lwp-request -useUS -m get http://www.eserve.com.sa/
[download]

The number of possible unknown errors is huge (goes into 11_000), to see some run

for my $i( 0 .. 256 ){ # 
    $! = $i;
    $^E = $i;
    printf "(%d)(%s)(%d)(%s)\n", $!,$!,$^E,$^E;
}
[download]

As for these redirected websites, the one in question http://www.eserve.com.sa is HTTP.

Which always redirects to the HTTPs version.

In any case, should I conclude that redirection isn't supported.

No.