Although correct, this is not the issue here (the process group id and the process id are numerically the same in this case).

The real issue is that by creating a new $fh with

    my $fh = new FileHandle();
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the old handle will go out of scope, so the file is automatically being closed.

If you store away $fh somewhere, e.g.

    my $fh = new FileHandle();
    push @fhandles, $fh;   # keep it from going out of scope
    ...
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the file handle count will go up.