Shortcode always at top of content

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You can’t just print output from a shortcode, you have to return it. Easiest way given your existing code is to use an output buffer. You can modify your draw_chain_menu() method like this:

//shortcode action
public function draw_chain_menu($args) {
    // previous code omitted...
    ob_start();
    $this->generate_tax_select($args);
    return ob_get_clean();
}