ke_search v3.4.2 produces invalide (x)html in TYPO3 v10.4.13

[al-Iskandar]

If I use ke_search v3.4.2 in Typo3 v10.4.13 ke_search produces in the results area a list of links with only the last item having a closing <li> tag:

<div id="kesearch_pagebrowser_bottom"> <div class="pages_total"> <div class="result_txt">Suchergebnisse 1 bis 10 von 548</div> <div class="kesearch_pagebrowser"> <ul> <li><a class="current" href="/xxx/suche/?tx_kesearch_pi1%5Bpage%5D=1&amp;cHash=xxx">1</a> <li><a href="/xxx/suche/?tx_kesearch_pi1%5Bpage%5D=2&amp;cHash=xxx">2</a> <li><a href="/xxx/suche/?tx_kesearch_pi1%5Bpage%5D=3&amp;cHash=xxx">3</a> <li><a href="/xxx/suche/?tx_kesearch_pi1%5Bpage%5D=4&amp;cHash=xxx">4</a> <li><a href="/xxx/suche/?tx_kesearch_pi1%5Bpage%5D=5&amp;cHash=xxx">5</a> <li><a href="/xxx/suche/?tx_kesearch_pi1%5Bpage%5D=6&amp;cHash=xxx">6</a> <li><a href="/xxx/suche/?tx_kesearch_pi1%5Bpage%5D=7&amp;cHash=xxx">7</a> <li><a href="/xxx/suche/?tx_kesearch_pi1%5Bpage%5D=8&amp;cHash=xxx">8</a> <li><a href="/xxx/suche/?tx_kesearch_pi1%5Bpage%5D=9&amp;cHash=xxx">9</a> <li><a href="/xxx/suche/?tx_kesearch_pi1%5Bpage%5D=10&amp;cHash=xxx">10</a> <li><a class="next" href="/xxxx/suche/?tx_kesearch_pi1%5Bpage%5D=2&amp;cHash=xxx">nächste</a></li> </ul> </div> </div>

PS: For usability in xhtml it would be nice, if the template would use   instead of