Navigation menu items do not appear after purging cache

I see this error when loading your page, though I don’t know if it’s relevant to your problem. When I first loaded the page, the background of your menu did not appear. When I refreshed, it did. You might check with tech support at Cloudflare.

A Parser-blocking, cross site (i.e. different eTLD+1) script, https://ajax.cloudflare.com/cdn-cgi/nexp/dok3v=f2befc48d1/cloudflare.min.js, is invoked via document.write. This MAY be blocked by the browser in this or a future page load due to poor network connectivity. If blocked in this page load, it will be confirmed in a subsequent console message.See https://www.chromestatus.com/feature/5718547946799104 for more details.

If you disable cloudflare (put it in development or pass-through mode), does the problem go away?

Try disabling W3TC, too.

You should ask Cloudflare.