Avoid the globs and use IO::File instead. Then everything will work fine.

while (<>){
  # do some stuff
  unless defined ( $fhs{$conn} )
  {
    $fhs{$conn} = IO::File->new();
    
    die $!
      unless $fhs{$conn}->open(">conn-$conn");
  }
  
  $fhs{$conn}->print ($_);
}
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Update: added code