Chrome Extension Manifest V3 Screen Recording Ends When Changing the Tab

I am trying to migrate my extension which records screen/tab/window according to the chosen option from manifest V2 to V3. In manifest V2 I was able to use background script as persistent and reach html page objects such as mediaRecorder, navigator. However in manifest V3 background script works as a service worker. So, I have to start the screen record in content-scripts to be able to reach the html objects. When I start chrome.desktopCapture API from the background script, I have to start the screenRecord in one of the tabs (should give a tabid to chrome.desktopCapture.chooseDesktopMedia API call). I cannot start it on the background page and when the page was refreshed or changed to a new URL screen record stops. Is there any workaround for this?