Opps sorry, i just noticed that, you have both Fancy Slider and Carousel extension.
And the carousel extension works okay while "fancy product" slider fail to work.
The "cache" issue is a known issue and has been fixed for Carousel Extension (it is valid what i have said in my previous post). So update the Carousel extension and the failure due to cache will be fixed (if you were experiencing it)
But if you are saying, fancy product slider fails to work when cache is enabled. Then you are wrong, because i am seeing a JAVA conflict issue in your website (right now) and this is why the Fancy Slider extension doesn't work. And this doesn't directly related to cache at all. (maybe cache is doing some other thing which leads to JAVA conflict though)
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The Conflict error you see in the screenshot: $.preloadImages is a function of our Fancy Slider extension, and it is a requirement for running.And the reason why fancy slider extension doesn't work is because of that conflict issue. Please revise your website's jQuery Library, and disable the jQuery duplication. 1 jQuery above all JAVA files will be enough for a single page, now i am seeing there are 2 jquery(s) get loaded in your website. One comes with widgetkit (/media/widgetkit/js/jquery.js), other one comes with mozalizer plugin (/plugins/system/modalizer/modals/jquery.min.js). This conflict might be related to this duplicated jQueries as well. I can't be sure though.
But first update the carousel extension with the newer version. Are you really going to use them together? Basically they are doing the same exact thing.... What happens when try to disable either one of them?
Regards.