I've just tested this here (I assume you mean the filter for New Posts?) and it was fine for me (changed the setting and saved, I even logged out as well as restarted my browser and it was still filtering with the new setting I had chosen).
Try deleting the cookie for this site and testing again. Occasionally I have had cookie issues on one or two (non-XenForo) sites at times with Chrome. Deleting the cookie and trying again resolved those problems (I guess the cookie was somehow corrupted).
This is all working as expected for me on multiple sites and apparently your defaults are set to {"started":true} which if I copy over to my own then it works as expected.