Do you have any prove that chmod and unlink were executed? Maybe $file_name has the wrong value, and so the '-e' test failed, and none of the if loop executed. I suggest expanding your test to:

if ( -e $file_name ) {
    say "Deleting $file_name."
    chmod 0777, $file_name or die "Could not delete $file_name: $!;
    unlink $file_name;
}
else {
    say "In the else  branch: '-e $file_name was false'.";
}
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Update - added missing $$$$