Turning off trim_urls makes sense. If your page is loading things
with ajax PageSpeed will be wrong about the context in which it thinks
the html will be shown, and it will trim urls incorrectly. This is
why the filter is off by default.
The other option is to use "pagespeed Disallow
*/path/to/broken/files/* to tell PageSpeed to ignore the requests that
it's mishandling.
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