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It seems you need a version of WhatsApp on your phone that isn't out yet.

Indeed. Play Store says I have the latest version... anybody have a working apk?

Try the following APK: http://www.apkmirror.com/apk/whatsapp-inc/whatsapp/whatsapp-.... (yes, it's not the newest release but they apparently did a rollback after this version removing the web menu item)