One thing you could try is enabling debugging in W3 Total Cache and view the page or post from a mobile device (or use the mobile view on your desktop system) and look at the generated HTML source and at the bottom, you should see a HTML comment with the W3 Total Cache debug info. This should tell you if the page is being cached or not and whether or not the cookie is being rejected.
That might provide some clues as to what's going on.
Peace...