Add a query_args cache context to the header block so that the cache is dependent on query string.

MODULENAME.module.php

<?php

function MODULENAME_block_build_alter(array &$build, \Drupal\Core\Block\BlockPluginInterface $block) {
  // Add the 'query args' cache context to the header block.
  if ($this_block_is_the_header_block) { // @TODO: Make this line work.
    $build['#cache']['contexts'][] = 'url.query_args';
  }
}