If I switch on Network Throttling in DevTools, and set it to Offline (as Lighthouse does), and I have a registered Service Worker with the following code:

self.onfetch = evt => {
  evt.respondWith(fetch(evt.request));
};

The fetch seems to succeed (assuming the device has a connection). This results in an uncanny valley, where the page is sort-of offline, but the SW isn't. 

While the solution is to disable connectivity at the OS level, it does make the 'offline' setting (or any other throttling) a misnomer if you have a SW. It would be great if either the UI indicated the scope of the throttling, or, if possible, we extended the throttling to cover SW-based fetches too.