##05Ab1E, 12 bytes##

    Dg0Qië€gÙg1Q

**Explained**

    Dg0Qi          # if empty list, return input
         ë         # else
          €g       # map length over list
            Ùg1Q   # check if length of uniquified list is 1

[Try it online](http://05ab1e.tryitonline.net/#code=RGcwUWnDq-KCrGfDmWcxUQ&input=W1sxXSwgWzFdLCBbMiwgM11d)