Uses the experimental Badging API to Badge messenger.com when installed as a desktop PWA.

messenger-unread-badges.js

// ==UserScript==
// @name Messenger.com Unread Count
// @namespace Violentmonkey Scripts
// @match https://www.messenger.com/*
// @grant none
// ==/UserScript==

const getUnreadCount = () => {
  return document.querySelectorAll('._1ht3').length;
}

const updateUnreadCount = () => {
  const Badge = navigator.setAppBadge || navigator.setExperimentalAppBadge;
  if (!Badge)
    return;
  
  const unreadCount = getUnreadCount();
  Badge.call(navigator, unreadCount);
}

const poll = (func, regularity) => {
  func();
  setTimeout(() => {
    poll(func, regularity);
  }, regularity)
};

poll(updateUnreadCount, 1000);