I don't know enough about Moose to explain why your attempt using DEMOLISH didn't work. I can suggest an alternative, though.

Another way would be to get a weakened reference to the circular chain.

my $weak_ref;

{
    my $y = Y->new();
    weaken( $weak_ref = $y );
    $y->{__CIRCULAR__} = $y if $ENV{TEST_THE_TEST};
}

ok(!defined($weak_ref), 'circular reference destruction');
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I think your method with the following tweak would be more reliable because it allows you to monitor all the objects, not just one. Change

sub { $garbage_collected = 1 }
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to

sub { ++$garbage_collected }
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Then you can construct a whole bunch of objects and make sure they all got destroyed.