Should be closed. Lexical variable falls out of scope, gets undefined, and resources assigned to it are reclaimed.

As an example:

package foo {
    sub DESTROY { warn "Destroying object" }
    sub new     {
        warn "Creating object";
        return bless {}, $_[0];
    }
}
warn "Before block";
{
    my $obj = foo->new;
}
warn "After block";
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The same basic concept lives around any lexical variable, including lexical file handles.