There is a big problem with auto-converting images, and its with HTTP caching.
Consider this: I have page.html, with an <img src=logo.jpg> tag. If I'm using Chrome and I request logo.jpg PageSpeed/the web server detects that I support WebP via the Accept header. It sends WebP image for the URL logo.jpg.
And that's a problem. Because the content for logo.jpg is no longer dependent entirely on the URL. It the URL, and the values of the User-Agent or the Accept headers, which determine the response.
This is called content negotiation and it utterly nullifies shared HTTP caching. This is bad because shared HTTP caching can be a huge performance win.
The right way to serve WebP images is to change the URL. The application logic that generates the HTML should detect that browser support webp and rewrite the image tag accordingly. In other words, page.html should include a <IMG src=logo.webp> tag.
This means that browsers that support webp get it, browsers that only support JPEG get it, and shared HTTP caches can be used without polluting the cache.
Sadly, PageSpeed and other technologies like it aren't currently smart enough to re-write the URL in the base HTML page. As always, technology that "automatically fixes problem X" rarely is the best solution.