If the main page content is a views page, panels everywhere fails to render the contextual links.
views_page_alter() populates $page['#contextual_links'] but theme_panels_everywhere_page() does not render it.

A workaround is to alter the page in template.php

/**
 * Implements hook_page_alter().
 */
function THEME_page_alter(&$page) {
  // Panels everywhere fails to render page contextual links.
  // Move them to $page['content'].
  if ($page['#theme'] === 'panels_everywhere_page' && !empty($page['#contextual_links'])) {
    $page['content']['#contextual_links'] = $page['#contextual_links'];
    unset($page['#contextual_links']);
  }
}

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ufku created an issue.